(* Title: HOL/Analysis/Convex_Euclidean_Space.thy
Author: L C Paulson, University of Cambridge
Author: Robert Himmelmann, TU Muenchen
Author: Bogdan Grechuk, University of Edinburgh
Author: Armin Heller, TU Muenchen
Author: Johannes Hoelzl, TU Muenchen
*)
section \<open>Convex Sets and Functions on (Normed) Euclidean Spaces\<close>
theory Convex_Euclidean_Space
imports
Convex
Topology_Euclidean_Space
begin
subsection\<^marker>\<open>tag unimportant\<close> \<open>Topological Properties of Convex Sets and Functions\<close>
lemma aff_dim_cball:
fixes a :: "'n::euclidean_space"
assumes "e > 0"
shows "aff_dim (cball a e) = int (DIM('n))"
proof -
have "(\x. a + x) ` (cball 0 e) \ cball a e"
unfolding cball_def dist_norm by auto
then have "aff_dim (cball (0 :: 'n::euclidean_space) e) \ aff_dim (cball a e)"
using aff_dim_translation_eq[of a "cball 0 e"]
aff_dim_subset[of "(+) a ` cball 0 e" "cball a e"]
by auto
moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
ultimately show ?thesis
using aff_dim_le_DIM[of "cball a e"] by auto
qed
lemma aff_dim_open:
fixes S :: "'n::euclidean_space set"
assumes "open S"
and "S \ {}"
shows "aff_dim S = int (DIM('n))"
proof -
obtain x where "x \ S"
using assms by auto
then obtain e where e: "e > 0" "cball x e \ S"
using open_contains_cball[of S] assms by auto
then have "aff_dim (cball x e) \ aff_dim S"
using aff_dim_subset by auto
with e show ?thesis
using aff_dim_cball[of e x] aff_dim_le_DIM[of S] by auto
qed
lemma low_dim_interior:
fixes S :: "'n::euclidean_space set"
assumes "\ aff_dim S = int (DIM('n))"
shows "interior S = {}"
proof -
have "aff_dim(interior S) \ aff_dim S"
using interior_subset aff_dim_subset[of "interior S" S] by auto
then show ?thesis
using aff_dim_open[of "interior S"] aff_dim_le_DIM[of S] assms by auto
qed
corollary empty_interior_lowdim:
fixes S :: "'n::euclidean_space set"
shows "dim S < DIM ('n) \ interior S = {}"
by (metis low_dim_interior affine_hull_UNIV dim_affine_hull less_not_refl dim_UNIV)
corollary aff_dim_nonempty_interior:
fixes S :: "'a::euclidean_space set"
shows "interior S \ {} \ aff_dim S = DIM('a)"
by (metis low_dim_interior)
subsection \<open>Relative interior of a set\<close>
definition\<^marker>\<open>tag important\<close> "rel_interior S =
{x. \<exists>T. openin (top_of_set (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
lemma rel_interior_mono:
"\S \ T; affine hull S = affine hull T\
\<Longrightarrow> (rel_interior S) \<subseteq> (rel_interior T)"
by (auto simp: rel_interior_def)
lemma rel_interior_maximal:
"\T \ S; openin(top_of_set (affine hull S)) T\ \ T \ (rel_interior S)"
by (auto simp: rel_interior_def)
lemma rel_interior: "rel_interior S = {x \ S. \T. open T \ x \ T \ T \ affine hull S \ S}"
(is "?lhs = ?rhs")
proof
show "?lhs \ ?rhs"
by (force simp add: rel_interior_def openin_open)
{ fix x T
assume *: "x \ S" "open T" "x \ T" "T \ affine hull S \ S"
then have **: "x \ T \ affine hull S"
using hull_inc by auto
with * have "\Tb. (\Ta. open Ta \ Tb = affine hull S \ Ta) \ x \ Tb \ Tb \ S"
by (rule_tac x = "T \ (affine hull S)" in exI) auto
}
then show "?rhs \ ?lhs"
by (force simp add: rel_interior_def openin_open)
qed
lemma mem_rel_interior: "x \ rel_interior S \ (\T. open T \ x \ T \ S \ T \ affine hull S \ S)"
by (auto simp: rel_interior)
lemma mem_rel_interior_ball:
"x \ rel_interior S \ x \ S \ (\e. e > 0 \ ball x e \ affine hull S \ S)"
(is "?lhs = ?rhs")
proof
assume ?rhs then show ?lhs
by (simp add: rel_interior) (meson Elementary_Metric_Spaces.open_ball centre_in_ball)
qed (force simp: rel_interior open_contains_ball)
lemma rel_interior_ball:
"rel_interior S = {x \ S. \e. e > 0 \ ball x e \ affine hull S \ S}"
using mem_rel_interior_ball [of _ S] by auto
lemma mem_rel_interior_cball:
"x \ rel_interior S \ x \ S \ (\e. e > 0 \ cball x e \ affine hull S \ S)"
(is "?lhs = ?rhs")
proof
assume ?rhs then obtain e where "x \ S" "e > 0" "cball x e \ affine hull S \ S"
by (auto simp: rel_interior)
then have "ball x e \ affine hull S \ S"
by auto
then show ?lhs
using \<open>0 < e\<close> \<open>x \<in> S\<close> rel_interior_ball by auto
qed (force simp: rel_interior open_contains_cball)
lemma rel_interior_cball:
"rel_interior S = {x \ S. \e. e > 0 \ cball x e \ affine hull S \ S}"
using mem_rel_interior_cball [of _ S] by auto
lemma rel_interior_empty [simp]: "rel_interior {} = {}"
by (auto simp: rel_interior_def)
lemma affine_hull_sing [simp]: "affine hull {a :: 'n::euclidean_space} = {a}"
by (metis affine_hull_eq affine_sing)
lemma rel_interior_sing [simp]:
fixes a :: "'n::euclidean_space" shows "rel_interior {a} = {a}"
proof -
have "\x::real. 0 < x"
using zero_less_one by blast
then show ?thesis
by (auto simp: rel_interior_ball)
qed
lemma subset_rel_interior:
fixes S T :: "'n::euclidean_space set"
assumes "S \ T"
and "affine hull S = affine hull T"
shows "rel_interior S \ rel_interior T"
using assms by (auto simp: rel_interior_def)
lemma rel_interior_subset: "rel_interior S \ S"
by (auto simp: rel_interior_def)
lemma rel_interior_subset_closure: "rel_interior S \ closure S"
using rel_interior_subset by (auto simp: closure_def)
lemma interior_subset_rel_interior: "interior S \ rel_interior S"
by (auto simp: rel_interior interior_def)
lemma interior_rel_interior:
fixes S :: "'n::euclidean_space set"
assumes "aff_dim S = int(DIM('n))"
shows "rel_interior S = interior S"
proof -
have "affine hull S = UNIV"
using assms affine_hull_UNIV[of S] by auto
then show ?thesis
unfolding rel_interior interior_def by auto
qed
lemma rel_interior_interior:
fixes S :: "'n::euclidean_space set"
assumes "affine hull S = UNIV"
shows "rel_interior S = interior S"
using assms unfolding rel_interior interior_def by auto
lemma rel_interior_open:
fixes S :: "'n::euclidean_space set"
assumes "open S"
shows "rel_interior S = S"
by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
lemma interior_rel_interior_gen:
fixes S :: "'n::euclidean_space set"
shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
by (metis interior_rel_interior low_dim_interior)
lemma rel_interior_nonempty_interior:
fixes S :: "'n::euclidean_space set"
shows "interior S \ {} \ rel_interior S = interior S"
by (metis interior_rel_interior_gen)
lemma affine_hull_nonempty_interior:
fixes S :: "'n::euclidean_space set"
shows "interior S \ {} \ affine hull S = UNIV"
by (metis affine_hull_UNIV interior_rel_interior_gen)
lemma rel_interior_affine_hull [simp]:
fixes S :: "'n::euclidean_space set"
shows "rel_interior (affine hull S) = affine hull S"
proof -
have *: "rel_interior (affine hull S) \ affine hull S"
using rel_interior_subset by auto
{
fix x
assume x: "x \ affine hull S"
define e :: real where "e = 1"
then have "e > 0" "ball x e \ affine hull (affine hull S) \ affine hull S"
using hull_hull[of _ S] by auto
then have "x \ rel_interior (affine hull S)"
using x rel_interior_ball[of "affine hull S"] by auto
}
then show ?thesis using * by auto
qed
lemma rel_interior_UNIV [simp]: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
by (metis open_UNIV rel_interior_open)
lemma rel_interior_convex_shrink:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "c \ rel_interior S"
and "x \ S"
and "0 < e"
and "e \ 1"
shows "x - e *\<^sub>R (x - c) \ rel_interior S"
proof -
obtain d where "d > 0" and d: "ball c d \ affine hull S \ S"
using assms(2) unfolding mem_rel_interior_ball by auto
{
fix y
assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d" "y \ affine hull S"
have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
using \<open>e > 0\<close> by (auto simp: scaleR_left_diff_distrib scaleR_right_diff_distrib)
have "x \ affine hull S"
using assms hull_subset[of S] by auto
moreover have "1 / e + - ((1 - e) / e) = 1"
using \<open>e > 0\<close> left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \ affine hull S"
using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"]
by (simp add: algebra_simps)
have "c - ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = (1 / e) *\<^sub>R (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
using \<open>e > 0\<close>
by (auto simp: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
then have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \1/e\ * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
unfolding dist_norm norm_scaleR[symmetric] by auto
also have "\ = \1/e\ * norm (x - e *\<^sub>R (x - c) - y)"
by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
also have "\ < d"
using as[unfolded dist_norm] and \<open>e > 0\<close>
by (auto simp:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
finally have "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \ S"
using "**" d by auto
then have "y \ S"
using * convexD [OF \<open>convex S\<close>] assms(3-5)
by (metis diff_add_cancel diff_ge_0_iff_ge le_add_same_cancel1 less_eq_real_def)
}
then have "ball (x - e *\<^sub>R (x - c)) (e*d) \ affine hull S \ S"
by auto
moreover have "e * d > 0"
using \<open>e > 0\<close> \<open>d > 0\<close> by simp
moreover have c: "c \ S"
using assms rel_interior_subset by auto
moreover from c have "x - e *\<^sub>R (x - c) \ S"
using convexD_alt[of S x c e] assms
by (metis diff_add_eq diff_diff_eq2 less_eq_real_def scaleR_diff_left scaleR_one scale_right_diff_distrib)
ultimately show ?thesis
using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] \e > 0\ by auto
qed
lemma interior_real_atLeast [simp]:
fixes a :: real
shows "interior {a..} = {a<..}"
proof -
{
fix y
have "ball y (y - a) \ {a..}"
by (auto simp: dist_norm)
moreover assume "a < y"
ultimately have "y \ interior {a..}"
by (force simp add: mem_interior)
}
moreover
{
fix y
assume "y \ interior {a..}"
then obtain e where e: "e > 0" "cball y e \ {a..}"
using mem_interior_cball[of y "{a..}"] by auto
moreover from e have "y - e \ cball y e"
by (auto simp: cball_def dist_norm)
ultimately have "a \ y - e" by blast
then have "a < y" using e by auto
}
ultimately show ?thesis by auto
qed
lemma continuous_ge_on_Ioo:
assumes "continuous_on {c..d} g" "\x. x \ {c<.. g x \ a" "c < d" "x \ {c..d}"
shows "g (x::real) \ (a::real)"
proof-
from assms(3) have "{c..d} = closure {c<.. by (rule closure_greaterThanLessThan[symmetric])
also from assms(2) have "{c<.. (g -` {a..} \ {c..d})" by auto
hence "closure {c<.. closure (g -` {a..} \ {c..d})" by (rule closure_mono)
also from assms(1) have "closed (g -` {a..} \ {c..d})"
by (auto simp: continuous_on_closed_vimage)
hence "closure (g -` {a..} \ {c..d}) = g -` {a..} \ {c..d}" by simp
finally show ?thesis using \<open>x \<in> {c..d}\<close> by auto
qed
lemma interior_real_atMost [simp]:
fixes a :: real
shows "interior {..a} = {..
proof -
{
fix y
have "ball y (a - y) \ {..a}"
by (auto simp: dist_norm)
moreover assume "a > y"
ultimately have "y \ interior {..a}"
by (force simp add: mem_interior)
}
moreover
{
fix y
assume "y \ interior {..a}"
then obtain e where e: "e > 0" "cball y e \ {..a}"
using mem_interior_cball[of y "{..a}"] by auto
moreover from e have "y + e \ cball y e"
by (auto simp: cball_def dist_norm)
ultimately have "a \ y + e" by auto
then have "a > y" using e by auto
}
ultimately show ?thesis by auto
qed
lemma interior_atLeastAtMost_real [simp]: "interior {a..b} = {a<..
proof-
have "{a..b} = {a..} \ {..b}" by auto
also have "interior \ = {a<..} \ {..
by (simp)
also have "\ = {a<..
finally show ?thesis .
qed
lemma interior_atLeastLessThan [simp]:
fixes a::real shows "interior {a..
by (metis atLeastLessThan_def greaterThanLessThan_def interior_atLeastAtMost_real interior_Int interior_interior interior_real_atLeast)
lemma interior_lessThanAtMost [simp]:
fixes a::real shows "interior {a<..b} = {a<..
by (metis atLeastAtMost_def greaterThanAtMost_def interior_atLeastAtMost_real interior_Int
interior_interior interior_real_atLeast)
lemma interior_greaterThanLessThan_real [simp]: "interior {a<..
by (metis interior_atLeastAtMost_real interior_interior)
lemma frontier_real_atMost [simp]:
fixes a :: real
shows "frontier {..a} = {a}"
unfolding frontier_def by auto
lemma frontier_real_atLeast [simp]: "frontier {a..} = {a::real}"
by (auto simp: frontier_def)
lemma frontier_real_greaterThan [simp]: "frontier {a<..} = {a::real}"
by (auto simp: interior_open frontier_def)
lemma frontier_real_lessThan [simp]: "frontier {..
by (auto simp: interior_open frontier_def)
lemma rel_interior_real_box [simp]:
fixes a b :: real
assumes "a < b"
shows "rel_interior {a .. b} = {a <..< b}"
proof -
have "box a b \ {}"
using assms
unfolding set_eq_iff
by (auto intro!: exI[of _ "(a + b) / 2"] simp: box_def)
then show ?thesis
using interior_rel_interior_gen[of "cbox a b", symmetric]
by (simp split: if_split_asm del: box_real add: box_real[symmetric])
qed
lemma rel_interior_real_semiline [simp]:
fixes a :: real
shows "rel_interior {a..} = {a<..}"
proof -
have *: "{a<..} \ {}"
unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
then show ?thesis using interior_real_atLeast interior_rel_interior_gen[of "{a..}"]
by (auto split: if_split_asm)
qed
subsubsection \<open>Relative open sets\<close>
definition\<^marker>\<open>tag important\<close> "rel_open S \<longleftrightarrow> rel_interior S = S"
lemma rel_open: "rel_open S \ openin (top_of_set (affine hull S)) S" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding rel_open_def rel_interior_def
using openin_subopen[of "top_of_set (affine hull S)" S] by auto
qed (auto simp: rel_open_def rel_interior_def)
lemma openin_rel_interior: "openin (top_of_set (affine hull S)) (rel_interior S)"
using openin_subopen by (fastforce simp add: rel_interior_def)
lemma openin_set_rel_interior:
"openin (top_of_set S) (rel_interior S)"
by (rule openin_subset_trans [OF openin_rel_interior rel_interior_subset hull_subset])
lemma affine_rel_open:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "rel_open S"
unfolding rel_open_def
using assms rel_interior_affine_hull[of S] affine_hull_eq[of S]
by metis
lemma affine_closed:
fixes S :: "'n::euclidean_space set"
assumes "affine S"
shows "closed S"
proof -
{
assume "S \ {}"
then obtain L where L: "subspace L" "affine_parallel S L"
using assms affine_parallel_subspace[of S] by auto
then obtain a where a: "S = ((+) a ` L)"
using affine_parallel_def[of L S] affine_parallel_commut by auto
from L have "closed L" using closed_subspace by auto
then have "closed S"
using closed_translation a by auto
}
then show ?thesis by auto
qed
lemma closure_affine_hull:
fixes S :: "'n::euclidean_space set"
shows "closure S \ affine hull S"
by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
lemma closed_affine_hull [iff]:
fixes S :: "'n::euclidean_space set"
shows "closed (affine hull S)"
by (metis affine_affine_hull affine_closed)
lemma closure_same_affine_hull [simp]:
fixes S :: "'n::euclidean_space set"
shows "affine hull (closure S) = affine hull S"
proof -
have "affine hull (closure S) \ affine hull S"
using hull_mono[of "closure S" "affine hull S" "affine"]
closure_affine_hull[of S] hull_hull[of "affine" S]
by auto
moreover have "affine hull (closure S) \ affine hull S"
using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
ultimately show ?thesis by auto
qed
lemma closure_aff_dim [simp]:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (closure S) = aff_dim S"
proof -
have "aff_dim S \ aff_dim (closure S)"
using aff_dim_subset closure_subset by auto
moreover have "aff_dim (closure S) \ aff_dim (affine hull S)"
using aff_dim_subset closure_affine_hull by blast
moreover have "aff_dim (affine hull S) = aff_dim S"
using aff_dim_affine_hull by auto
ultimately show ?thesis by auto
qed
lemma rel_interior_closure_convex_shrink:
fixes S :: "_::euclidean_space set"
assumes "convex S"
and "c \ rel_interior S"
and "x \ closure S"
and "e > 0"
and "e \ 1"
shows "x - e *\<^sub>R (x - c) \ rel_interior S"
proof -
obtain d where "d > 0" and d: "ball c d \ affine hull S \ S"
using assms(2) unfolding mem_rel_interior_ball by auto
have "\y \ S. norm (y - x) * (1 - e) < e * d"
proof (cases "x \ S")
case True
then show ?thesis using \<open>e > 0\<close> \<open>d > 0\<close> by force
next
case False
then have x: "x islimpt S"
using assms(3)[unfolded closure_def] by auto
show ?thesis
proof (cases "e = 1")
case True
obtain y where "y \ S" "y \ x" "dist y x < 1"
using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
then show ?thesis
unfolding True using \<open>d > 0\<close> by (force simp add: )
next
case False
then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
then obtain y where "y \ S" "y \ x" "dist y x < e * d / (1 - e)"
using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
then show ?thesis
unfolding dist_norm using pos_less_divide_eq[OF *] by force
qed
qed
then obtain y where "y \ S" and y: "norm (y - x) * (1 - e) < e * d"
by auto
define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
unfolding z_def using \<open>e > 0\<close>
by (auto simp: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
have zball: "z \ ball c d"
using mem_ball z_def dist_norm[of c]
using y and assms(4,5)
by (simp add: norm_minus_commute) (simp add: field_simps)
have "x \ affine hull S"
using closure_affine_hull assms by auto
moreover have "y \ affine hull S"
using \<open>y \<in> S\<close> hull_subset[of S] by auto
moreover have "c \ affine hull S"
using assms rel_interior_subset hull_subset[of S] by auto
ultimately have "z \ affine hull S"
using z_def affine_affine_hull[of S]
mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
assms
by simp
then have "z \ S" using d zball by auto
obtain d1 where "d1 > 0" and d1: "ball z d1 \ ball c d"
using zball open_ball[of c d] openE[of "ball c d" z] by auto
then have "ball z d1 \ affine hull S \ ball c d \ affine hull S"
by auto
then have "ball z d1 \ affine hull S \ S"
using d by auto
then have "z \ rel_interior S"
using mem_rel_interior_ball using \<open>d1 > 0\<close> \<open>z \<in> S\<close> by auto
then have "y - e *\<^sub>R (y - z) \ rel_interior S"
using rel_interior_convex_shrink[of S z y e] assms \<open>y \<in> S\<close> by auto
then show ?thesis using * by auto
qed
lemma rel_interior_eq:
"rel_interior s = s \ openin(top_of_set (affine hull s)) s"
using rel_open rel_open_def by blast
lemma rel_interior_openin:
"openin(top_of_set (affine hull s)) s \ rel_interior s = s"
by (simp add: rel_interior_eq)
lemma rel_interior_affine:
fixes S :: "'n::euclidean_space set"
shows "affine S \ rel_interior S = S"
using affine_rel_open rel_open_def by auto
lemma rel_interior_eq_closure:
fixes S :: "'n::euclidean_space set"
shows "rel_interior S = closure S \ affine S"
proof (cases "S = {}")
case True
then show ?thesis
by auto
next
case False show ?thesis
proof
assume eq: "rel_interior S = closure S"
have "openin (top_of_set (affine hull S)) S"
by (metis eq closure_subset openin_rel_interior rel_interior_subset subset_antisym)
moreover have "closedin (top_of_set (affine hull S)) S"
by (metis closed_subset closure_subset_eq eq hull_subset rel_interior_subset)
ultimately have "S = {} \ S = affine hull S"
using convex_connected connected_clopen convex_affine_hull by metis
with False have "affine hull S = S"
by auto
then show "affine S"
by (metis affine_hull_eq)
next
assume "affine S"
then show "rel_interior S = closure S"
by (simp add: rel_interior_affine affine_closed)
qed
qed
subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Relative interior preserves under linear transformations\<close>
lemma rel_interior_translation_aux:
fixes a :: "'n::euclidean_space"
shows "((\x. a + x) ` rel_interior S) \ rel_interior ((\x. a + x) ` S)"
proof -
{
fix x
assume x: "x \ rel_interior S"
then obtain T where "open T" "x \ T \ S" "T \ affine hull S \ S"
using mem_rel_interior[of x S] by auto
then have "open ((\x. a + x) ` T)"
and "a + x \ ((\x. a + x) ` T) \ ((\x. a + x) ` S)"
and "((\x. a + x) ` T) \ affine hull ((\x. a + x) ` S) \ (\x. a + x) ` S"
using affine_hull_translation[of a S] open_translation[of T a] x by auto
then have "a + x \ rel_interior ((\x. a + x) ` S)"
using mem_rel_interior[of "a+x" "((\x. a + x) ` S)"] by auto
}
then show ?thesis by auto
qed
lemma rel_interior_translation:
fixes a :: "'n::euclidean_space"
shows "rel_interior ((\x. a + x) ` S) = (\x. a + x) ` rel_interior S"
proof -
have "(\x. (-a) + x) ` rel_interior ((\x. a + x) ` S) \ rel_interior S"
using rel_interior_translation_aux[of "-a" "(\x. a + x) ` S"]
translation_assoc[of "-a" "a"]
by auto
then have "((\x. a + x) ` rel_interior S) \ rel_interior ((\x. a + x) ` S)"
using translation_inverse_subset[of a "rel_interior ((+) a ` S)" "rel_interior S"]
by auto
then show ?thesis
using rel_interior_translation_aux[of a S] by auto
qed
lemma affine_hull_linear_image:
assumes "bounded_linear f"
shows "f ` (affine hull s) = affine hull f ` s"
proof -
interpret f: bounded_linear f by fact
have "affine {x. f x \ affine hull f ` s}"
unfolding affine_def
by (auto simp: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format])
moreover have "affine {x. x \ f ` (affine hull s)}"
using affine_affine_hull[unfolded affine_def, of s]
unfolding affine_def by (auto simp: f.scaleR [symmetric] f.add [symmetric])
ultimately show ?thesis
by (auto simp: hull_inc elim!: hull_induct)
qed
lemma rel_interior_injective_on_span_linear_image:
fixes f :: "'m::euclidean_space \ 'n::euclidean_space"
and S :: "'m::euclidean_space set"
assumes "bounded_linear f"
and "inj_on f (span S)"
shows "rel_interior (f ` S) = f ` (rel_interior S)"
proof -
{
fix z
assume z: "z \ rel_interior (f ` S)"
then have "z \ f ` S"
using rel_interior_subset[of "f ` S"] by auto
then obtain x where x: "x \ S" "f x = z" by auto
obtain e2 where e2: "e2 > 0" "cball z e2 \ affine hull (f ` S) \ (f ` S)"
using z rel_interior_cball[of "f ` S"] by auto
obtain K where K: "K > 0" "\x. norm (f x) \ norm x * K"
using assms Real_Vector_Spaces.bounded_linear.pos_bounded[of f] by auto
define e1 where "e1 = 1 / K"
then have e1: "e1 > 0" "\x. e1 * norm (f x) \ norm x"
using K pos_le_divide_eq[of e1] by auto
define e where "e = e1 * e2"
then have "e > 0" using e1 e2 by auto
{
fix y
assume y: "y \ cball x e \ affine hull S"
then have h1: "f y \ affine hull (f ` S)"
using affine_hull_linear_image[of f S] assms by auto
from y have "norm (x-y) \ e1 * e2"
using cball_def[of x e] dist_norm[of x y] e_def by auto
moreover have "f x - f y = f (x - y)"
using assms linear_diff[of f x y] linear_conv_bounded_linear[of f] by auto
moreover have "e1 * norm (f (x-y)) \ norm (x - y)"
using e1 by auto
ultimately have "e1 * norm ((f x)-(f y)) \ e1 * e2"
by auto
then have "f y \ cball z e2"
using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1 x by auto
then have "f y \ f ` S"
using y e2 h1 by auto
then have "y \ S"
using assms y hull_subset[of S] affine_hull_subset_span
inj_on_image_mem_iff [OF \<open>inj_on f (span S)\<close>]
by (metis Int_iff span_superset subsetCE)
}
then have "z \ f ` (rel_interior S)"
using mem_rel_interior_cball[of x S] \<open>e > 0\<close> x by auto
}
moreover
{
fix x
assume x: "x \ rel_interior S"
then obtain e2 where e2: "e2 > 0" "cball x e2 \ affine hull S \ S"
using rel_interior_cball[of S] by auto
have "x \ S" using x rel_interior_subset by auto
then have *: "f x \ f ` S" by auto
have "\x\span S. f x = 0 \ x = 0"
using assms subspace_span linear_conv_bounded_linear[of f]
linear_injective_on_subspace_0[of f "span S"]
by auto
then obtain e1 where e1: "e1 > 0" "\x \ span S. e1 * norm x \ norm (f x)"
using assms injective_imp_isometric[of "span S" f]
subspace_span[of S] closed_subspace[of "span S"]
by auto
define e where "e = e1 * e2"
hence "e > 0" using e1 e2 by auto
{
fix y
assume y: "y \ cball (f x) e \ affine hull (f ` S)"
then have "y \ f ` (affine hull S)"
using affine_hull_linear_image[of f S] assms by auto
then obtain xy where xy: "xy \ affine hull S" "f xy = y" by auto
with y have "norm (f x - f xy) \ e1 * e2"
using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
moreover have "f x - f xy = f (x - xy)"
using assms linear_diff[of f x xy] linear_conv_bounded_linear[of f] by auto
moreover have *: "x - xy \ span S"
using subspace_diff[of "span S" x xy] subspace_span \<open>x \<in> S\<close> xy
affine_hull_subset_span[of S] span_superset
by auto
moreover from * have "e1 * norm (x - xy) \ norm (f (x - xy))"
using e1 by auto
ultimately have "e1 * norm (x - xy) \ e1 * e2"
by auto
then have "xy \ cball x e2"
using cball_def[of x e2] dist_norm[of x xy] e1 by auto
then have "y \ f ` S"
using xy e2 by auto
}
then have "f x \ rel_interior (f ` S)"
using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * \<open>e > 0\<close> by auto
}
ultimately show ?thesis by auto
qed
lemma rel_interior_injective_linear_image:
fixes f :: "'m::euclidean_space \ 'n::euclidean_space"
assumes "bounded_linear f"
and "inj f"
shows "rel_interior (f ` S) = f ` (rel_interior S)"
using assms rel_interior_injective_on_span_linear_image[of f S]
subset_inj_on[of f "UNIV" "span S"]
by auto
subsection\<^marker>\<open>tag unimportant\<close> \<open>Openness and compactness are preserved by convex hull operation\<close>
lemma open_convex_hull[intro]:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open (convex hull S)"
proof (clarsimp simp: open_contains_cball convex_hull_explicit)
fix T and u :: "'a\real"
assume obt: "finite T" "T\S" "\x\T. 0 \ u x" "sum u T = 1"
from assms[unfolded open_contains_cball] obtain b
where b: "\x. x\S \ 0 < b x \ cball x (b x) \ S" by metis
have "b ` T \ {}"
using obt by auto
define i where "i = b ` T"
let ?\<Phi> = "\<lambda>y. \<exists>F. finite F \<and> F \<subseteq> S \<and> (\<exists>u. (\<forall>x\<in>F. 0 \<le> u x) \<and> sum u F = 1 \<and> (\<Sum>v\<in>F. u v *\<^sub>R v) = y)"
let ?a = "\v\T. u v *\<^sub>R v"
show "\e > 0. cball ?a e \ {y. ?\ y}"
proof (intro exI subsetI conjI)
show "0 < Min i"
unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] \<open>b ` T\<noteq>{}\<close>]
using b \<open>T\<subseteq>S\<close> by auto
next
fix y
assume "y \ cball ?a (Min i)"
then have y: "norm (?a - y) \ Min i"
unfolding dist_norm[symmetric] by auto
{ fix x
assume "x \ T"
then have "Min i \ b x"
by (simp add: i_def obt(1))
then have "x + (y - ?a) \ cball x (b x)"
using y unfolding mem_cball dist_norm by auto
moreover have "x \ S"
using \<open>x\<in>T\<close> \<open>T\<subseteq>S\<close> by auto
ultimately have "x + (y - ?a) \ S"
using y b by blast
}
moreover
have *: "inj_on (\v. v + (y - ?a)) T"
unfolding inj_on_def by auto
have "(\v\(\v. v + (y - ?a)) ` T. u (v - (y - ?a)) *\<^sub>R v) = y"
unfolding sum.reindex[OF *] o_def using obt(4)
by (simp add: sum.distrib sum_subtractf scaleR_left.sum[symmetric] scaleR_right_distrib)
ultimately show "y \ {y. ?\ y}"
proof (intro CollectI exI conjI)
show "finite ((\v. v + (y - ?a)) ` T)"
by (simp add: obt(1))
show "sum (\v. u (v - (y - ?a))) ((\v. v + (y - ?a)) ` T) = 1"
unfolding sum.reindex[OF *] o_def using obt(4) by auto
qed (use obt(1, 3) in auto)
qed
qed
lemma compact_convex_combinations:
fixes S T :: "'a::real_normed_vector set"
assumes "compact S" "compact T"
shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \ u \ u \ 1 \ x \ S \ y \ T}"
proof -
let ?X = "{0..1} \ S \ T"
let ?h = "(\z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
have *: "{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \ u \ u \ 1 \ x \ S \ y \ T} = ?h ` ?X"
by force
have "continuous_on ?X (\z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
unfolding continuous_on by (rule ballI) (intro tendsto_intros)
with assms show ?thesis
by (simp add: * compact_Times compact_continuous_image)
qed
lemma finite_imp_compact_convex_hull:
fixes S :: "'a::real_normed_vector set"
assumes "finite S"
shows "compact (convex hull S)"
proof (cases "S = {}")
case True
then show ?thesis by simp
next
case False
with assms show ?thesis
proof (induct rule: finite_ne_induct)
case (singleton x)
show ?case by simp
next
case (insert x A)
let ?f = "\(u, y::'a). u *\<^sub>R x + (1 - u) *\<^sub>R y"
let ?T = "{0..1::real} \ (convex hull A)"
have "continuous_on ?T ?f"
unfolding split_def continuous_on by (intro ballI tendsto_intros)
moreover have "compact ?T"
by (intro compact_Times compact_Icc insert)
ultimately have "compact (?f ` ?T)"
by (rule compact_continuous_image)
also have "?f ` ?T = convex hull (insert x A)"
unfolding convex_hull_insert [OF \<open>A \<noteq> {}\<close>]
apply safe
apply (rule_tac x=a in exI, simp)
apply (rule_tac x="1 - a" in exI, simp, fast)
apply (rule_tac x="(u, b)" in image_eqI, simp_all)
done
finally show "compact (convex hull (insert x A))" .
qed
qed
lemma compact_convex_hull:
fixes S :: "'a::euclidean_space set"
assumes "compact S"
shows "compact (convex hull S)"
proof (cases "S = {}")
case True
then show ?thesis using compact_empty by simp
next
case False
then obtain w where "w \ S" by auto
show ?thesis
unfolding caratheodory[of S]
proof (induct ("DIM('a) + 1"))
case 0
have *: "{x.\sa. finite sa \ sa \ S \ card sa \ 0 \ x \ convex hull sa} = {}"
using compact_empty by auto
from 0 show ?case unfolding * by simp
next
case (Suc n)
show ?case
proof (cases "n = 0")
case True
have "{x. \T. finite T \ T \ S \ card T \ Suc n \ x \ convex hull T} = S"
unfolding set_eq_iff and mem_Collect_eq
proof (rule, rule)
fix x
assume "\T. finite T \ T \ S \ card T \ Suc n \ x \ convex hull T"
then obtain T where T: "finite T" "T \ S" "card T \ Suc n" "x \ convex hull T"
by auto
show "x \ S"
proof (cases "card T = 0")
case True
then show ?thesis
using T(4) unfolding card_0_eq[OF T(1)] by simp
next
case False
then have "card T = Suc 0" using T(3) \<open>n=0\<close> by auto
then obtain a where "T = {a}" unfolding card_Suc_eq by auto
then show ?thesis using T(2,4) by simp
qed
next
fix x assume "x\S"
then show "\T. finite T \ T \ S \ card T \ Suc n \ x \ convex hull T"
by (rule_tac x="{x}" in exI) (use convex_hull_singleton in auto)
qed
then show ?thesis using assms by simp
next
case False
have "{x. \T. finite T \ T \ S \ card T \ Suc n \ x \ convex hull T} =
{(1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> {x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> x \<in> convex hull T}}"
unfolding set_eq_iff and mem_Collect_eq
proof (rule, rule)
fix x
assume "\u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \
0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
then obtain u v c T where obt: "x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
"0 \ c \ c \ 1" "u \ S" "finite T" "T \ S" "card T \ n" "v \ convex hull T"
by auto
moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \ convex hull insert u T"
by (meson convexD_alt convex_convex_hull hull_inc hull_mono in_mono insertCI obt(2) obt(7) subset_insertI)
ultimately show "\T. finite T \ T \ S \ card T \ Suc n \ x \ convex hull T"
by (rule_tac x="insert u T" in exI) (auto simp: card_insert_if)
next
fix x
assume "\T. finite T \ T \ S \ card T \ Suc n \ x \ convex hull T"
then obtain T where T: "finite T" "T \ S" "card T \ Suc n" "x \ convex hull T"
by auto
show "\u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \
0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
proof (cases "card T = Suc n")
case False
then have "card T \ n" using T(3) by auto
then show ?thesis
using \<open>w\<in>S\<close> and T
by (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) auto
next
case True
then obtain a u where au: "T = insert a u" "a\u"
by (metis card_le_Suc_iff order_refl)
show ?thesis
proof (cases "u = {}")
case True
then have "x = a" using T(4)[unfolded au] by auto
show ?thesis unfolding \<open>x = a\<close>
using T \<open>n \<noteq> 0\<close> unfolding au
by (rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI) force
next
case False
obtain ux vx b where obt: "ux\0" "vx\0" "ux + vx = 1"
"b \ convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
using T(4)[unfolded au convex_hull_insert[OF False]]
by auto
have *: "1 - vx = ux" using obt(3) by auto
show ?thesis
using obt T(1-3) card_insert_disjoint[OF _ au(2)] unfolding au *
by (rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI) force
qed
qed
qed
then show ?thesis
using compact_convex_combinations[OF assms Suc] by simp
qed
qed
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Extremal points of a simplex are some vertices\<close>
lemma dist_increases_online:
fixes a b d :: "'a::real_inner"
assumes "d \ 0"
shows "dist a (b + d) > dist a b \ dist a (b - d) > dist a b"
proof (cases "inner a d - inner b d > 0")
case True
then have "0 < inner d d + (inner a d * 2 - inner b d * 2)"
using assms
by (intro add_pos_pos) auto
then show ?thesis
unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
by (simp add: algebra_simps inner_commute)
next
case False
then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
using assms
by (intro add_pos_nonneg) auto
then show ?thesis
unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
by (simp add: algebra_simps inner_commute)
qed
lemma norm_increases_online:
fixes d :: "'a::real_inner"
shows "d \ 0 \ norm (a + d) > norm a \ norm(a - d) > norm a"
using dist_increases_online[of d a 0] unfolding dist_norm by auto
lemma simplex_furthest_lt:
fixes S :: "'a::real_inner set"
assumes "finite S"
shows "\x \ convex hull S. x \ S \ (\y \ convex hull S. norm (x - a) < norm(y - a))"
using assms
proof induct
fix x S
assume as: "finite S" "x\S" "\x\convex hull S. x \ S \ (\y\convex hull S. norm (x - a) < norm (y - a))"
show "\xa\convex hull insert x S. xa \ insert x S \
(\<exists>y\<in>convex hull insert x S. norm (xa - a) < norm (y - a))"
proof (intro impI ballI, cases "S = {}")
case False
fix y
assume y: "y \ convex hull insert x S" "y \ insert x S"
obtain u v b where obt: "u\0" "v\0" "u + v = 1" "b \ convex hull S" "y = u *\<^sub>R x + v *\<^sub>R b"
using y(1)[unfolded convex_hull_insert[OF False]] by auto
show "\z\convex hull insert x S. norm (y - a) < norm (z - a)"
proof (cases "y \ convex hull S")
case True
then obtain z where "z \ convex hull S" "norm (y - a) < norm (z - a)"
using as(3)[THEN bspec[where x=y]] and y(2) by auto
then show ?thesis
by (meson hull_mono subsetD subset_insertI)
next
case False
show ?thesis
proof (cases "u = 0 \ v = 0")
case True
with False show ?thesis
using obt y by auto
next
case False
then obtain w where w: "w>0" "w "w
using field_lbound_gt_zero[of u v] and obt(1,2) by auto
have "x \ b"
proof
assume "x = b"
then have "y = b" unfolding obt(5)
using obt(3) by (auto simp: scaleR_left_distrib[symmetric])
then show False using obt(4) and False
using \<open>x = b\<close> y(2) by blast
qed
then have *: "w *\<^sub>R (x - b) \ 0" using w(1) by auto
show ?thesis
using dist_increases_online[OF *, of a y]
proof (elim disjE)
assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
then have "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
unfolding dist_commute[of a]
unfolding dist_norm obt(5)
by (simp add: algebra_simps)
moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \ convex hull insert x S"
unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
proof (intro CollectI conjI exI)
show "u + w \ 0" "v - w \ 0"
using obt(1) w by auto
qed (use obt in auto)
ultimately show ?thesis by auto
next
assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
then have "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
unfolding dist_commute[of a]
unfolding dist_norm obt(5)
by (simp add: algebra_simps)
moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \ convex hull insert x S"
unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
proof (intro CollectI conjI exI)
show "u - w \ 0" "v + w \ 0"
using obt(1) w by auto
qed (use obt in auto)
ultimately show ?thesis by auto
qed
qed
qed
qed auto
qed (auto simp: assms)
lemma simplex_furthest_le:
fixes S :: "'a::real_inner set"
assumes "finite S"
and "S \ {}"
shows "\y\S. \x\ convex hull S. norm (x - a) \ norm (y - a)"
proof -
have "convex hull S \ {}"
using hull_subset[of S convex] and assms(2) by auto
then obtain x where x: "x \ convex hull S" "\y\convex hull S. norm (y - a) \ norm (x - a)"
using distance_attains_sup[OF finite_imp_compact_convex_hull[OF \<open>finite S\<close>], of a]
unfolding dist_commute[of a]
unfolding dist_norm
by auto
show ?thesis
proof (cases "x \ S")
case False
then obtain y where "y \ convex hull S" "norm (x - a) < norm (y - a)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1)
by auto
then show ?thesis
using x(2)[THEN bspec[where x=y]] by auto
next
case True
with x show ?thesis by auto
qed
qed
lemma simplex_furthest_le_exists:
fixes S :: "('a::real_inner) set"
shows "finite S \ \x\(convex hull S). \y\S. norm (x - a) \ norm (y - a)"
using simplex_furthest_le[of S] by (cases "S = {}") auto
lemma simplex_extremal_le:
fixes S :: "'a::real_inner set"
assumes "finite S"
and "S \ {}"
shows "\u\S. \v\S. \x\convex hull S. \y \ convex hull S. norm (x - y) \ norm (u - v)"
proof -
have "convex hull S \ {}"
using hull_subset[of S convex] and assms(2) by auto
then obtain u v where obt: "u \ convex hull S" "v \ convex hull S"
"\x\convex hull S. \y\convex hull S. norm (x - y) \ norm (u - v)"
using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]]
by (auto simp: dist_norm)
then show ?thesis
proof (cases "u\S \ v\S", elim disjE)
assume "u \ S"
then obtain y where "y \ convex hull S" "norm (u - v) < norm (y - v)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1)
by auto
then show ?thesis
using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2)
by auto
next
assume "v \ S"
then obtain y where "y \ convex hull S" "norm (v - u) < norm (y - u)"
using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2)
by auto
then show ?thesis
using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
by (auto simp: norm_minus_commute)
qed auto
qed
lemma simplex_extremal_le_exists:
fixes S :: "'a::real_inner set"
shows "finite S \ x \ convex hull S \ y \ convex hull S \
\<exists>u\<in>S. \<exists>v\<in>S. norm (x - y) \<le> norm (u - v)"
using convex_hull_empty simplex_extremal_le[of S]
by(cases "S = {}") auto
subsection \<open>Closest point of a convex set is unique, with a continuous projection\<close>
definition\<^marker>\<open>tag important\<close> closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a"
where "closest_point S a = (SOME x. x \ S \ (\y\S. dist a x \ dist a y))"
lemma closest_point_exists:
assumes "closed S"
and "S \ {}"
shows closest_point_in_set: "closest_point S a \ S"
and "\y\S. dist a (closest_point S a) \ dist a y"
unfolding closest_point_def
by (rule_tac someI2_ex, auto intro: distance_attains_inf[OF assms(1,2), of a])+
lemma closest_point_le: "closed S \ x \ S \ dist a (closest_point S a) \ dist a x"
using closest_point_exists[of S] by auto
lemma closest_point_self:
assumes "x \ S"
shows "closest_point S x = x"
unfolding closest_point_def
by (rule some1_equality, rule ex1I[of _ x]) (use assms in auto)
lemma closest_point_refl: "closed S \ S \ {} \ closest_point S x = x \ x \ S"
using closest_point_in_set[of S x] closest_point_self[of x S]
by auto
lemma closer_points_lemma:
assumes "inner y z > 0"
shows "\u>0. \v>0. v \ u \ norm(v *\<^sub>R z - y) < norm y"
proof -
have z: "inner z z > 0"
unfolding inner_gt_zero_iff using assms by auto
have "norm (v *\<^sub>R z - y) < norm y"
if "0 < v" and "v \ inner y z / inner z z" for v
unfolding norm_lt using z assms that
by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ \<open>0<v\<close>])
then show ?thesis
using assms z
by (rule_tac x = "inner y z / inner z z" in exI) auto
qed
lemma closer_point_lemma:
assumes "inner (y - x) (z - x) > 0"
shows "\u>0. u \ 1 \ dist (x + u *\<^sub>R (z - x)) y < dist x y"
proof -
obtain u where "u > 0"
and u: "\v. \0 u\ \ norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
using closer_points_lemma[OF assms] by auto
show ?thesis
using u[of "min u 1"] and \<open>u > 0\<close>
by (metis diff_diff_add dist_commute dist_norm less_eq_real_def not_less u zero_less_one)
qed
lemma any_closest_point_dot:
assumes "convex S" "closed S" "x \ S" "y \ S" "\z\S. dist a x \ dist a z"
shows "inner (a - x) (y - x) \ 0"
proof (rule ccontr)
assume "\ ?thesis"
then obtain u where u: "u>0" "u\1" "dist (x + u *\<^sub>R (y - x)) a < dist x a"
using closer_point_lemma[of a x y] by auto
let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y"
have "?z \ S"
using convexD_alt[OF assms(1,3,4), of u] using u by auto
then show False
using assms(5)[THEN bspec[where x="?z"]] and u(3)
by (auto simp: dist_commute algebra_simps)
qed
lemma any_closest_point_unique:
fixes x :: "'a::real_inner"
assumes "convex S" "closed S" "x \ S" "y \ S"
"\z\S. dist a x \ dist a z" "\z\S. dist a y \ dist a z"
shows "x = y"
using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
unfolding norm_pths(1) and norm_le_square
by (auto simp: algebra_simps)
lemma closest_point_unique:
assumes "convex S" "closed S" "x \ S" "\z\S. dist a x \ dist a z"
shows "x = closest_point S a"
using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point S a"]
using closest_point_exists[OF assms(2)] and assms(3) by auto
lemma closest_point_dot:
assumes "convex S" "closed S" "x \ S"
shows "inner (a - closest_point S a) (x - closest_point S a) \ 0"
using any_closest_point_dot[OF assms(1,2) _ assms(3)]
by (metis assms(2) assms(3) closest_point_in_set closest_point_le empty_iff)
lemma closest_point_lt:
assumes "convex S" "closed S" "x \ S" "x \ closest_point S a"
shows "dist a (closest_point S a) < dist a x"
using closest_point_unique[where a=a] closest_point_le[where a=a] assms by fastforce
lemma setdist_closest_point:
"\closed S; S \ {}\ \ setdist {a} S = dist a (closest_point S a)"
by (metis closest_point_exists(2) closest_point_in_set emptyE insert_iff setdist_unique)
lemma closest_point_lipschitz:
assumes "convex S"
and "closed S" "S \ {}"
shows "dist (closest_point S x) (closest_point S y) \ dist x y"
proof -
have "inner (x - closest_point S x) (closest_point S y - closest_point S x) \ 0"
and "inner (y - closest_point S y) (closest_point S x - closest_point S y) \ 0"
by (simp_all add: assms closest_point_dot closest_point_in_set)
then show ?thesis unfolding dist_norm and norm_le
using inner_ge_zero[of "(x - closest_point S x) - (y - closest_point S y)"]
by (simp add: inner_add inner_diff inner_commute)
qed
lemma continuous_at_closest_point:
assumes "convex S"
and "closed S"
and "S \ {}"
shows "continuous (at x) (closest_point S)"
unfolding continuous_at_eps_delta
using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
lemma continuous_on_closest_point:
assumes "convex S"
and "closed S"
and "S \ {}"
shows "continuous_on t (closest_point S)"
by (metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
proposition closest_point_in_rel_interior:
assumes "closed S" "S \ {}" and x: "x \ affine hull S"
shows "closest_point S x \ rel_interior S \ x \ rel_interior S"
proof (cases "x \ S")
case True
then show ?thesis
by (simp add: closest_point_self)
next
case False
then have "False" if asm: "closest_point S x \ rel_interior S"
proof -
obtain e where "e > 0" and clox: "closest_point S x \ S"
and e: "cball (closest_point S x) e \ affine hull S \ S"
using asm mem_rel_interior_cball by blast
then have clo_notx: "closest_point S x \ x"
using \<open>x \<notin> S\<close> by auto
define y where "y \ closest_point S x -
(min 1 (e / norm(closest_point S x - x))) *\<^sub>R (closest_point S x - x)"
have "x - y = (1 - min 1 (e / norm (closest_point S x - x))) *\<^sub>R (x - closest_point S x)"
by (simp add: y_def algebra_simps)
then have "norm (x - y) = abs ((1 - min 1 (e / norm (closest_point S x - x)))) * norm(x - closest_point S x)"
by simp
also have "\ < norm(x - closest_point S x)"
using clo_notx \<open>e > 0\<close>
by (auto simp: mult_less_cancel_right2 field_split_simps)
finally have no_less: "norm (x - y) < norm (x - closest_point S x)" .
have "y \ affine hull S"
unfolding y_def
by (meson affine_affine_hull clox hull_subset mem_affine_3_minus2 subsetD x)
moreover have "dist (closest_point S x) y \ e"
using \<open>e > 0\<close> by (auto simp: y_def min_mult_distrib_right)
ultimately have "y \ S"
using subsetD [OF e] by simp
then have "dist x (closest_point S x) \ dist x y"
by (simp add: closest_point_le \<open>closed S\<close>)
with no_less show False
by (simp add: dist_norm)
qed
moreover have "x \ rel_interior S"
using rel_interior_subset False by blast
ultimately show ?thesis by blast
qed
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Various point-to-set separating/supporting hyperplane theorems\<close>
lemma supporting_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex S"
and "closed S"
and "S \ {}"
and "z \ S"
shows "\a b. \y\S. inner a z < b \ inner a y = b \ (\x\S. inner a x \ b)"
proof -
obtain y where "y \ S" and y: "\x\S. dist z y \ dist z x"
by (metis distance_attains_inf[OF assms(2-3)])
show ?thesis
proof (intro exI bexI conjI ballI)
show "(y - z) \ z < (y - z) \ y"
by (metis \<open>y \<in> S\<close> assms(4) diff_gt_0_iff_gt inner_commute inner_diff_left inner_gt_zero_iff right_minus_eq)
show "(y - z) \ y \ (y - z) \ x" if "x \ S" for x
proof (rule ccontr)
have *: "\u. 0 \ u \ u \ 1 \ dist z y \ dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
using assms(1)[unfolded convex_alt] and y and \<open>x\<in>S\<close> and \<open>y\<in>S\<close> by auto
assume "\ (y - z) \ y \ (y - z) \ x"
then obtain v where "v > 0" "v \ 1" "dist (y + v *\<^sub>R (x - y)) z < dist y z"
using closer_point_lemma[of z y x] by (auto simp: inner_diff)
then show False
using *[of v] by (auto simp: dist_commute algebra_simps)
qed
qed (use \<open>y \<in> S\<close> in auto)
qed
lemma separating_hyperplane_closed_point:
fixes z :: "'a::{real_inner,heine_borel}"
assumes "convex S"
and "closed S"
and "z \ S"
shows "\a b. inner a z < b \ (\x\S. inner a x > b)"
proof (cases "S = {}")
case True
then show ?thesis
by (simp add: gt_ex)
next
case False
obtain y where "y \ S" and y: "\x. x \ S \ dist z y \ dist z x"
by (metis distance_attains_inf[OF assms(2) False])
show ?thesis
proof (intro exI conjI ballI)
show "(y - z) \ z < inner (y - z) z + (norm (y - z))\<^sup>2 / 2"
using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
next
fix x
assume "x \ S"
have "False" if *: "0 < inner (z - y) (x - y)"
proof -
obtain u where "u > 0" "u \ 1" "dist (y + u *\<^sub>R (x - y)) z < dist y z"
using * closer_point_lemma by blast
then show False using y[of "y + u *\<^sub>R (x - y)"] convexD_alt [OF \convex S\]
using \<open>x\<in>S\<close> \<open>y\<in>S\<close> by (auto simp: dist_commute algebra_simps)
qed
moreover have "0 < (norm (y - z))\<^sup>2"
using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
then have "0 < inner (y - z) (y - z)"
unfolding power2_norm_eq_inner by simp
ultimately show "(y - z) \ z + (norm (y - z))\<^sup>2 / 2 < (y - z) \ x"
by (force simp: field_simps power2_norm_eq_inner inner_commute inner_diff)
qed
qed
lemma separating_hyperplane_closed_0:
assumes "convex (S::('a::euclidean_space) set)"
and "closed S"
and "0 \ S"
shows "\a b. a \ 0 \ 0 < b \ (\x\S. inner a x > b)"
proof (cases "S = {}")
case True
have "(SOME i. i\Basis) \ (0::'a)"
by (metis Basis_zero SOME_Basis)
then show ?thesis
using True zero_less_one by blast
next
case False
then show ?thesis
using False using separating_hyperplane_closed_point[OF assms]
by (metis all_not_in_conv inner_zero_left inner_zero_right less_eq_real_def not_le)
qed
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Now set-to-set for closed/compact sets\<close>
lemma separating_hyperplane_closed_compact:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "closed S"
and "convex T"
and "compact T"
and "T \ {}"
and "S \ T = {}"
shows "\a b. (\x\S. inner a x < b) \ (\x\T. inner a x > b)"
proof (cases "S = {}")
case True
obtain b where b: "b > 0" "\x\T. norm x \ b"
using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
obtain z :: 'a where z: "norm z = b + 1"
using vector_choose_size[of "b + 1"] and b(1) by auto
then have "z \ T" using b(2)[THEN bspec[where x=z]] by auto
then obtain a b where ab: "inner a z < b" "\x\T. b < inner a x"
using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z]
by auto
then show ?thesis
using True by auto
next
case False
then obtain y where "y \ S" by auto
obtain a b where "0 < b" and \<section>: "\<And>x. x \<in> (\<Union>x\<in> S. \<Union>y \<in> T. {x - y}) \<Longrightarrow> b < inner a x"
using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
using closed_compact_differences assms by fastforce
have ab: "b + inner a y < inner a x" if "x\S" "y\T" for x y
using \<section> [of "x-y"] that by (auto simp add: inner_diff_right less_diff_eq)
define k where "k = (SUP x\T. a \ x)"
have "k + b / 2 < a \ x" if "x \ S" for x
proof -
have "k \ inner a x - b"
unfolding k_def
using \<open>T \<noteq> {}\<close> ab that by (fastforce intro: cSUP_least)
then show ?thesis
using \<open>0 < b\<close> by auto
qed
moreover
have "- (k + b / 2) < - a \ x" if "x \ T" for x
proof -
have "inner a x - b / 2 < k"
unfolding k_def
proof (subst less_cSUP_iff)
show "T \ {}" by fact
show "bdd_above ((\) a ` T)"
using ab[rule_format, of y] \<open>y \<in> S\<close>
by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
show "\y\T. a \ x - b / 2 < a \ y"
using \<open>0 < b\<close> that by force
qed
then show ?thesis
by auto
qed
ultimately show ?thesis
by (metis inner_minus_left neg_less_iff_less)
qed
lemma separating_hyperplane_compact_closed:
fixes S :: "'a::euclidean_space set"
assumes "convex S"
and "compact S"
and "S \ {}"
and "convex T"
and "closed T"
and "S \ T = {}"
shows "\a b. (\x\S. inner a x < b) \ (\x\T. inner a x > b)"
proof -
obtain a b where "(\x\T. inner a x < b) \ (\x\S. b < inner a x)"
by (metis disjoint_iff_not_equal separating_hyperplane_closed_compact assms)
then show ?thesis
by (metis inner_minus_left neg_less_iff_less)
qed
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>General case without assuming closure and getting non-strict separation\<close>
lemma separating_hyperplane_set_0:
assumes "convex S" "(0::'a::euclidean_space) \ S"
shows "\a. a \ 0 \ (\x\S. 0 \ inner a x)"
proof -
let ?k = "\c. {x::'a. 0 \ inner c x}"
have *: "frontier (cball 0 1) \ \f \ {}" if as: "f \ ?k ` S" "finite f" for f
proof -
obtain c where c: "f = ?k ` c" "c \ S" "finite c"
using finite_subset_image[OF as(2,1)] by auto
then obtain a b where ab: "a \ 0" "0 < b" "\x\convex hull c. b < inner a x"
using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
using subset_hull[of convex, OF assms(1), symmetric, of c]
by force
have "norm (a /\<^sub>R norm a) = 1"
by (simp add: ab(1))
moreover have "(\y\c. 0 \ y \ (a /\<^sub>R norm a))"
using hull_subset[of c convex] ab by (force simp: inner_commute)
ultimately have "\x. norm x = 1 \ (\y\c. 0 \ inner y x)"
by blast
then show "frontier (cball 0 1) \ \f \ {}"
unfolding c(1) frontier_cball sphere_def dist_norm by auto
qed
have "frontier (cball 0 1) \ (\(?k ` S)) \ {}"
by (rule compact_imp_fip) (use * closed_halfspace_ge in auto)
then obtain x where "norm x = 1" "\y\S. x\?k y"
unfolding frontier_cball dist_norm sphere_def by auto
then show ?thesis
by (metis inner_commute mem_Collect_eq norm_eq_zero zero_neq_one)
qed
lemma separating_hyperplane_sets:
fixes S T :: "'a::euclidean_space set"
assumes "convex S"
and "convex T"
and "S \ {}"
and "T \ {}"
and "S \ T = {}"
shows "\a b. a \ 0 \ (\x\S. inner a x \ b) \ (\x\T. inner a x \ b)"
proof -
from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
obtain a where "a \ 0" "\x\{x - y |x y. x \ T \ y \ S}. 0 \ inner a x"
using assms(3-5) by force
then have *: "\x y. x \ T \ y \ S \ inner a y \ inner a x"
by (force simp: inner_diff)
then have bdd: "bdd_above (((\) a)`S)"
using \<open>T \<noteq> {}\<close> by (auto intro: bdd_aboveI2[OF *])
show ?thesis
using \<open>a\<noteq>0\<close>
by (intro exI[of _ a] exI[of _ "SUP x\S. a \ x"])
(auto intro!: cSUP_upper bdd cSUP_least \<open>a \<noteq> 0\<close> \<open>S \<noteq> {}\<close> *)
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>More convexity generalities\<close>
lemma convex_closure [intro,simp]:
fixes S :: "'a::real_normed_vector set"
assumes "convex S"
shows "convex (closure S)"
apply (rule convexI)
unfolding closure_sequential
apply (elim exE)
subgoal for x y u v f g
by (rule_tac x="\n. u *\<^sub>R f n + v *\<^sub>R g n" in exI) (force intro: tendsto_intros dest: convexD [OF assms])
done
lemma convex_interior [intro,simp]:
fixes S :: "'a::real_normed_vector set"
assumes "convex S"
shows "convex (interior S)"
unfolding convex_alt Ball_def mem_interior
proof clarify
fix x y u
assume u: "0 \ u" "u \ (1::real)"
fix e d
assume ed: "ball x e \ S" "ball y d \ S" "0
show "\e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \ S"
proof (intro exI conjI subsetI)
fix z
assume z: "z \ ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
have "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \ S"
proof (rule_tac assms[unfolded convex_alt, rule_format])
show "z - u *\<^sub>R (y - x) \ S" "z + (1 - u) *\<^sub>R (y - x) \ S"
using ed z u by (auto simp add: algebra_simps dist_norm)
qed (use u in auto)
then show "z \ S"
using u by (auto simp: algebra_simps)
qed(use u ed in auto)
qed
lemma convex_hull_eq_empty[simp]: "convex hull S = {} \ S = {}"
using hull_subset[of S convex] convex_hull_empty by auto
subsection\<^marker>\<open>tag unimportant\<close> \<open>Convex set as intersection of halfspaces\<close>
lemma convex_halfspace_intersection:
fixes S :: "('a::euclidean_space) set"
assumes "closed S" "convex S"
shows "S = \{h. S \ h \ (\a b. h = {x. inner a x \ b})}"
proof -
{ fix z
assume "\T. S \ T \ (\a b. T = {x. inner a x \ b}) \ z \ T" "z \ S"
then have \<section>: "\<And>a b. S \<subseteq> {x. inner a x \<le> b} \<Longrightarrow> z \<in> {x. inner a x \<le> b}"
by blast
obtain a b where "inner a z < b" "(\x\S. inner a x > b)"
using \<open>z \<notin> S\<close> assms separating_hyperplane_closed_point by blast
then have False
--> --------------------
--> maximum size reached
--> --------------------
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