Theory Starlike

(* Title:      HOL/Analysis/Starlike.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
*)
chapter ‹Unsorted›

theory Starlike
  imports
    Convex_Euclidean_Space
    Line_Segment
begin

lemma affine_hull_closed_segment [simp]:
     "affine hull (closed_segment a b) = affine hull {a,b}"
  by (simp add: segment_convex_hull)

lemma affine_hull_open_segment [simp]:
    fixes a :: "'a::euclidean_space"
    shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)

lemma rel_interior_closure_convex_segment:
  fixes S :: "_::euclidean_space set"
  assumes "convex S" "a  rel_interior S" "b  closure S"
    shows "open_segment a b  rel_interior S"
proof
  fix x
  have [simp]: "(1 - u) *R a + u *R b = b - (1 - u) *R (b - a)" for u
    by (simp add: algebra_simps)
  assume "x  open_segment a b"
  then show "x  rel_interior S"
    unfolding closed_segment_def open_segment_def  using assms
    by (auto intro: rel_interior_closure_convex_shrink)
qed

lemma convex_hull_insert_segments:
   "convex hull (insert a S) =
    (if S = {} then {a} else  x  convex hull S. closed_segment a x)"
  by (force simp add: convex_hull_insert_alt in_segment)

lemma Int_convex_hull_insert_rel_exterior:
  fixes z :: "'a::euclidean_space"
  assumes "convex C" "T  C" and z: "z  rel_interior C" and dis: "disjnt S (rel_interior C)"
  shows "S  (convex hull (insert z T)) = S  (convex hull T)" (is "?lhs = ?rhs")
proof
  have "T = {}  z  S"
    using dis z by (auto simp add: disjnt_def)
  then show "?lhs  ?rhs"
  proof (clarsimp simp add: convex_hull_insert_segments)
    fix x y
    assume "x  S" and y: "y  convex hull T" and "x  closed_segment z y"
    have "y  closure C"
      by (metis y convex C T  C closure_subset contra_subsetD convex_hull_eq hull_mono)
    moreover have "x  rel_interior C"
      by (meson x  S dis disjnt_iff)
    moreover have "x  open_segment z y  {z, y}"
      using x  closed_segment z y closed_segment_eq_open by blast
    ultimately show "x  convex hull T"
      using rel_interior_closure_convex_segment [OF convex C z]
      using y z by blast
  qed
  show "?rhs  ?lhs"
    by (meson hull_mono inf_mono subset_insertI subset_refl)
qed

subsectiontag unimportant› ‹Shrinking towards the interior of a convex set›

lemma mem_interior_convex_shrink:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S"
    and "c  interior S"
    and "x  S"
    and "0 < e"
    and "e  1"
  shows "x - e *R (x - c)  interior S"
proof -
  obtain d where "d > 0" and d: "ball c d  S"
    using assms(2) unfolding mem_interior by auto
  show ?thesis
    unfolding mem_interior
  proof (intro exI subsetI conjI)
    fix y
    assume "y  ball (x - e *R (x - c)) (e*d)"
    then have as: "dist (x - e *R (x - c)) y < e * d"
      by simp
    have *: "y = (1 - (1 - e)) *R ((1 / e) *R y - ((1 - e) / e) *R x) + (1 - e) *R x"
      using e > 0 by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
    have "c - ((1 / e) *R y - ((1 - e) / e) *R x) = (1 / e) *R (e *R c - y + (1 - e) *R x)"
      using e > 0
      by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
    then have "dist c ((1 / e) *R y - ((1 - e) / e) *R x) = ¦1/e¦ * norm (e *R c - y + (1 - e) *R x)"
      by (simp add: dist_norm)
    also have " = ¦1/e¦ * norm (x - e *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 e > 0
      by (auto simp add:pos_divide_less_eq[OF e > 0] mult.commute)
    finally have "(1 - (1 - e)) *R ((1 / e) *R y - ((1 - e) / e) *R x) + (1 - e) *R x  S"
      using assms(3-5) d
      by (intro convexD_alt [OF convex S]) (auto intro: convexD_alt [OF convex S])
    with e > 0 show "y  S"
      by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
  qed (use e>0 d>0 in auto)
qed

lemma mem_interior_closure_convex_shrink:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S"
    and "c  interior S"
    and "x  closure S"
    and "0 < e"
    and "e  1"
  shows "x - e *R (x - c)  interior S"
proof -
  obtain d where "d > 0" and d: "ball c d  S"
    using assms(2) unfolding mem_interior by auto
  have "yS. norm (y - x) * (1 - e) < e * d"
  proof (cases "x  S")
    case True
    then show ?thesis
      using e > 0 d > 0 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
        using True 0 < d by auto
    next
      case False
      then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
        using e  1 e > 0 d > 0 by auto
      then obtain y where "y  S" "y  x" "dist y x < e * d / (1 - e)"
        using islimpt_approachable x by blast
      then have "norm (y - x) * (1 - e) < e * d"
        by (metis "*" dist_norm mult_imp_div_pos_le not_less)
      then show ?thesis
        using y  S by blast
    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) *R (x - y)"
  have *: "x - e *R (x - c) = y - e *R (y - z)"
    unfolding z_def using e > 0
    by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
  have "(1 - e) * norm (x - y) / e < d"
    using y 0 < e by (simp add: field_simps norm_minus_commute)
  then have "z  interior (ball c d)"
    using 0 < e e  1 by (simp add: interior_open[OF open_ball] z_def dist_norm)
  then have "z  interior S"
    using d interiorI interior_ball by blast
  then show ?thesis
    unfolding * using mem_interior_convex_shrink y  S assms by blast
qed

lemma in_interior_closure_convex_segment:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" and a: "a  interior S" and b: "b  closure S"
    shows "open_segment a b  interior S"
proof (clarsimp simp: in_segment)
  fix u::real
  assume u: "0 < u" "u < 1"
  have "(1 - u) *R a + u *R b = b - (1 - u) *R (b - a)"
    by (simp add: algebra_simps)
  also have "...  interior S" using mem_interior_closure_convex_shrink [OF assms] u
    by simp
  finally show "(1 - u) *R a + u *R b  interior S" .
qed

lemma convex_closure_interior:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" and int: "interior S  {}"
  shows "closure(interior S) = closure S"
proof -
  obtain a where a: "a  interior S"
    using int by auto
  have "closure S  closure(interior S)"
  proof
    fix x
    assume x: "x  closure S"
    show "x  closure (interior S)"
    proof (cases "x=a")
      case True
      then show ?thesis
        using a  interior S closure_subset by blast
    next
      case False
      show ?thesis
      proof (clarsimp simp add: closure_def islimpt_approachable)
        fix e::real
        assume xnotS: "x  interior S" and "0 < e"
        show "x'interior S. x'  x  dist x' x < e"
        proof (intro bexI conjI)
          show "x - min (e/2 / norm (x - a)) 1 *R (x - a)  x"
            using False 0 < e by (auto simp: algebra_simps min_def)
          show "dist (x - min (e/2 / norm (x - a)) 1 *R (x - a)) x < e"
            using 0 < e by (auto simp: dist_norm min_def)
          show "x - min (e/2 / norm (x - a)) 1 *R (x - a)  interior S"
            using 0 < e False
            by (auto simp add: min_def a intro: mem_interior_closure_convex_shrink [OF convex S a x])
        qed
      qed
    qed
  qed
  then show ?thesis
    by (simp add: closure_mono interior_subset subset_antisym)
qed

lemma closure_convex_Int_superset:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "interior S  {}" "interior S  closure T"
  shows "closure(S  T) = closure S"
proof -
  have "closure S  closure(interior S)"
    by (simp add: convex_closure_interior assms)
  also have "...  closure (S  T)"
    using interior_subset [of S] assms
    by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
  finally show ?thesis
    by (simp add: closure_mono dual_order.antisym)
qed


subsectiontag unimportant› ‹Some obvious but surprisingly hard simplex lemmas›

lemma simplex:
  assumes "finite S"
    and "0  S"
  shows "convex hull (insert 0 S) = {y. u. (xS. 0  u x)  sum u S  1  sum (λx. u x *R x) S = y}"
proof (simp add: convex_hull_finite set_eq_iff assms, safe)
  fix x and u :: "'a  real"
  assume "0  u 0" "xS. 0  u x" "u 0 + sum u S = 1"
  then show "v. (xS. 0  v x)  sum v S  1  (xS. v x *R x) = (xS. u x *R x)"
    by force
next
  fix x and u :: "'a  real"
  assume "xS. 0  u x" "sum u S  1"
  then show "v. 0  v 0  (xS. 0  v x)  v 0 + sum v S = 1  (xS. v x *R x) = (xS. u x *R x)"
    by (rule_tac x="λx. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
qed

lemma substd_simplex:
  assumes d: "d  Basis"
  shows "convex hull (insert 0 d) =
    {x. (iBasis. 0  xi)  (id. xi)  1  (iBasis. i  d  xi = 0)}"
  (is "convex hull (insert 0 ?p) = ?s")
proof -
  let ?D = d
  have "0  ?p"
    using assms by (auto simp: image_def)
  from d have "finite d"
    by (blast intro: finite_subset finite_Basis)
  show ?thesis
    unfolding simplex[OF finite d 0  ?p]
  proof (intro set_eqI; safe)
    fix u :: "'a  real"
    assume as: "x?D. 0  u x" "sum u ?D  1" 
    let ?x = "(x?D. u x *R x)"
    have ind: "iBasis. i  d  u i = ?x  i"
      and notind: "(iBasis. i  d  ?x  i = 0)"
      using substdbasis_expansion_unique[OF assms] by blast+
    then have **: "sum u ?D = sum ((∙) ?x) ?D"
      using assms by (auto intro!: sum.cong)
    show "0  ?x  i" if "i  Basis" for i
      using as(1) ind notind that by fastforce
    show "sum ((∙) ?x) ?D  1"
      using "**" as(2) by linarith
    show "?x  i = 0" if "i  Basis" "i  d" for i
      using notind that by blast
  next
    fix x 
    assume "iBasis. 0  x  i" "sum ((∙) x) ?D  1" "(iBasis. i  d  x  i = 0)"
    with d show "u. (x?D. 0  u x)  sum u ?D  1  (x?D. u x *R x) = x"
      unfolding substdbasis_expansion_unique[OF assms] 
      by (rule_tac x="inner x" in exI) auto
  qed
qed

lemma std_simplex:
  "convex hull (insert 0 Basis) =
    {x::'a::euclidean_space. (iBasis. 0  xi)  sum (λi. xi) Basis  1}"
  using substd_simplex[of Basis] by auto

lemma interior_std_simplex:
  "interior (convex hull (insert 0 Basis)) =
    {x::'a::euclidean_space. (iBasis. 0 < xi)  sum (λi. xi) Basis < 1}"
  unfolding set_eq_iff mem_interior std_simplex
proof (intro allI iffI CollectI; clarify)
  fix x :: 'a
  fix e
  assume "e > 0" and as: "ball x e  {x. (iBasis. 0  x  i)  sum ((∙) x) Basis  1}"
  show "(iBasis. 0 < x  i)  sum ((∙) x) Basis < 1"
  proof safe
    fix i :: 'a
    assume i: "i  Basis"
    then show "0 < x  i"
      using as[THEN subsetD[where c="x - (e/2) *R i"]] and e > 0 
      by (force simp add: inner_simps)
  next
    have **: "dist x (x + (e/2) *R (SOME i. iBasis)) < e" using e > 0
      unfolding dist_norm
      by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
    have "i. i  Basis  (x + (e/2) *R (SOME i. iBasis))  i =
      xi + (if i = (SOME i. iBasis) then e/2 else 0)"
      by (auto simp: SOME_Basis inner_Basis inner_simps)
    then have *: "sum ((∙) (x + (e/2) *R (SOME i. iBasis))) Basis =
      sum (λi. xi + (if (SOME i. iBasis) = i then e/2 else 0)) Basis"
      by (auto simp: intro!: sum.cong)
    have "sum ((∙) x) Basis < sum ((∙) (x + (e/2) *R (SOME i. iBasis))) Basis"
      using e > 0 DIM_positive by (auto simp: SOME_Basis sum.distrib *)
    also have "  1"
      using ** as by force
    finally show "sum ((∙) x) Basis < 1" by auto
  qed 
next
  fix x :: 'a
  assume as: "iBasis. 0 < x  i" "sum ((∙) x) Basis < 1"
  obtain a :: 'b where "a  UNIV" using UNIV_witness ..
  let ?d = "(1 - sum ((∙) x) Basis) / real (DIM('a))"
  show "e>0. ball x e  {x. (iBasis. 0  x  i)  sum ((∙) x) Basis  1}"
  proof (rule_tac x="min (Min (((∙) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
    fix y
    assume y: "y  ball x (min (Min ((∙) x ` Basis)) ?d)"
    have "sum ((∙) y) Basis  sum (λi. xi + ?d) Basis"
    proof (rule sum_mono)
      fix i :: 'a
      assume i: "i  Basis"
      have "¦yi - xi¦  norm (y - x)"
        by (metis Basis_le_norm i inner_commute inner_diff_right)
      also have "... < ?d"
        using y by (simp add: dist_norm norm_minus_commute)
      finally have "¦yi - xi¦ < ?d" .
      then show "y  i  x  i + ?d" by auto
    qed
    also have "  1"
      unfolding sum.distrib sum_constant
      by (auto simp add: Suc_le_eq)
    finally show "sum ((∙) y) Basis  1" .
    show "(iBasis. 0  y  i)"
    proof safe
      fix i :: 'a
      assume i: "i  Basis"
      have "norm (x - y) < Min (((∙) x) ` Basis)"
        using y by (auto simp: dist_norm less_eq_real_def)
      also have "...  xi"
        using i by auto
      finally have "norm (x - y) < xi" .
      then show "0  yi"
        using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
        by (auto simp: inner_simps)
    qed
  next
    have "Min (((∙) x) ` Basis) > 0"
      using as by simp
    moreover have "?d > 0"
      using as by (auto simp: Suc_le_eq)
    ultimately show "0 < min (Min ((∙) x ` Basis)) ((1 - sum ((∙) x) Basis) / real DIM('a))"
      by linarith
  qed 
qed

lemma interior_std_simplex_nonempty:
  obtains a :: "'a::euclidean_space" where
    "a  interior(convex hull (insert 0 Basis))"
proof -
  let ?D = "Basis :: 'a set"
  let ?a = "sum (λb::'a. inverse (2 * real DIM('a)) *R b) Basis"
  {
    fix i :: 'a
    assume i: "i  Basis"
    have "?a  i = inverse (2 * real DIM('a))"
      by (rule trans[of _ "sum (λj. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
         (simp_all add: sum.If_cases i) }
  note ** = this
  show ?thesis
  proof
    show "?a  interior(convex hull (insert 0 Basis))"
      unfolding interior_std_simplex mem_Collect_eq
    proof safe
      fix i :: 'a
      assume i: "i  Basis"
      show "0 < ?a  i"
        unfolding **[OF i] by (auto simp add: Suc_le_eq)
    next
      have "sum ((∙) ?a) ?D = sum (λi. inverse (2 * real DIM('a))) ?D"
        by (auto intro: sum.cong)
      also have " < 1"
        unfolding sum_constant divide_inverse[symmetric]
        by (auto simp add: field_simps)
      finally show "sum ((∙) ?a) ?D < 1" by auto
    qed
  qed
qed

lemma rel_interior_substd_simplex:
  assumes D: "D  Basis"
  shows "rel_interior (convex hull (insert 0 D)) =
         {x::'a::euclidean_space. (iD. 0 < xi)  (iD. xi) < 1  (iBasis. i  D  xi = 0)}"
     (is "_ = ?s")
proof -
  have "finite D"
    using D finite_Basis finite_subset by blast
  show ?thesis
  proof (cases "D = {}")
    case True
    then show ?thesis
      using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
  next
    case False
    have h0: "affine hull (convex hull (insert 0 D)) =
              {x::'a::euclidean_space. (iBasis. i  D  xi = 0)}"
      using affine_hull_convex_hull affine_hull_substd_basis assms by auto
    have aux: "x::'a. iBasis. (iD. 0  xi)  (iBasis. i  D  xi = 0)  0  xi"
      by auto
    {
      fix x :: "'a::euclidean_space"
      assume x: "x  rel_interior (convex hull (insert 0 D))"
      then obtain e where "e > 0" and
        "ball x e  {xa. (iBasis. i  D  xai = 0)}  convex hull (insert 0 D)"
        using mem_rel_interior_ball[of x "convex hull (insert 0 D)"] h0 by auto
      then have as: "y. dist x y < e  (iBasis. i  D  yi = 0) 
                            (iD. 0  y  i)  sum ((∙) y) D  1"
        using assms by (force simp: substd_simplex)
      have x0: "(iBasis. i  D  xi = 0)"
        using x rel_interior_subset  substd_simplex[OF assms] by auto
      have "(iD. 0 < x  i)  sum ((∙) x) D < 1  (iBasis. i  D  xi = 0)"
      proof (intro conjI ballI)
        fix i :: 'a
        assume "i  D"
        then have "jD. 0  (x - (e/2) *R i)  j"
          using D e > 0 x0
          by (intro as[THEN conjunct1]) (force simp: dist_norm inner_simps inner_Basis)
        then show "0 < x  i"
          using e > 0 i  D D  by (force simp: inner_simps inner_Basis)
      next
        obtain a where a: "a  D"
          using D  {} by auto
        then have **: "dist x (x + (e/2) *R a) < e"
          using e > 0 norm_Basis[of a] D by (auto simp: dist_norm)
        have "i. i  Basis  (x + (e/2) *R a)  i = xi + (if i = a then e/2 else 0)"
          using a D by (auto simp: inner_simps inner_Basis)
        then have *: "sum ((∙) (x + (e/2) *R a)) D = sum (λi. xi + (if a = i then e/2 else 0)) D"
          using D by (intro sum.cong) auto
        have "a  Basis"
          using a  D D by auto
        then have h1: "(iBasis. i  D  (x + (e/2) *R a)  i = 0)"
          using x0 D aD by (auto simp add: inner_add_left inner_Basis)
        have "sum ((∙) x) D < sum ((∙) (x + (e/2) *R a)) D"
          using e > 0 a  D finite D by (auto simp add: * sum.distrib)
        also have "  1"
          using ** h1 as[rule_format, of "x + (e/2) *R a"]
          by auto
        finally show "sum ((∙) x) D < 1" "i. iBasis  i  D  xi = 0"
          using x0 by auto
      qed
    }
    moreover
    {
      fix x :: "'a::euclidean_space"
      assume as: "x  ?s"
      have "i. 0 < xi  0 = xi  0  xi"
        by auto
      moreover have "i. i  D  i  D" by auto
      ultimately
      have "i. (iD. 0 < xi)  (i. i  D  xi = 0)  0  xi"
        by metis
      then have h2: "x  convex hull (insert 0 D)"
        using as assms by (force simp add: substd_simplex)
      obtain a where a: "a  D"
        using D  {} by auto
      define d where "d  (1 - sum ((∙) x) D) / real (card D)"
      have "e>0. ball x e  {x. iBasis. i  D  x  i = 0}  convex hull insert 0 D"
        unfolding substd_simplex[OF assms]
      proof (intro exI; safe)
        have "0 < card D" using D  {} finite D
          by (simp add: card_gt_0_iff)
        have "Min (((∙) x) ` D) > 0"
          using as D  {} finite D by (simp)
        moreover have "d > 0" 
          using as 0 < card D by (auto simp: d_def)
        ultimately show "min (Min (((∙) x) ` D)) d > 0"
          by auto
        fix y :: 'a
        assume y2: "iBasis. i  D  yi = 0"
        assume "y  ball x (min (Min ((∙) x ` D)) d)"
        then have y: "dist x y < min (Min ((∙) x ` D)) d"
          by auto
        have "sum ((∙) y) D  sum (λi. xi + d) D"
        proof (rule sum_mono)
          fix i
          assume "i  D"
          with D have i: "i  Basis"
            by auto
          have "¦yi - xi¦  norm (y - x)"
            by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
          also have "... < d"
            by (metis dist_norm min_less_iff_conj norm_minus_commute y)
          finally have "¦yi - xi¦ < d" .
          then show "y  i  x  i + d" by auto
        qed
        also have "  1"
          unfolding sum.distrib sum_constant d_def using 0 < card D
          by auto
        finally show "sum ((∙) y) D  1" .

        fix i :: 'a
        assume i: "i  Basis"
        then show "0  yi"
        proof (cases "iD")
          case True
          have "norm (x - y) < xi"
            using y Min_gr_iff[of "(∙) x ` D" "norm (x - y)"] 0 < card D i  D
            by (simp add: dist_norm card_gt_0_iff)
          then show "0  yi"
            using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
            by (auto simp: inner_simps)
        qed (use y2 in auto)
      qed
      then have "x  rel_interior (convex hull (insert 0 D))"
        using h0 h2 rel_interior_ball by force
    }
    ultimately have
      "x. x  rel_interior (convex hull insert 0 D) 
        x  {x. (iD. 0 < x  i)  sum ((∙) x) D < 1  (iBasis. i  D  x  i = 0)}"
      by blast
    then show ?thesis by (rule set_eqI)
  qed
qed

lemma rel_interior_substd_simplex_nonempty:
  assumes "D  {}"
    and "D  Basis"
  obtains a :: "'a::euclidean_space"
    where "a  rel_interior (convex hull (insert 0 D))"
proof -
  let ?a = "sum (λb::'a::euclidean_space. inverse (2 * real (card D)) *R b) D"
  have "finite D"
    using assms finite_Basis infinite_super by blast
  then have d1: "0 < real (card D)"
    using D  {} by auto
  {
    fix i
    assume "i  D"
    have "?a  i = sum (λj. if i = j then inverse (2 * real (card D)) else 0) D"
      unfolding inner_sum_left
      using i  D by (auto simp: inner_Basis subsetD[OF assms(2)] intro: sum.cong)
    also have "... = inverse (2 * real (card D))"
      using i  D finite D by auto
    finally have "?a  i = inverse (2 * real (card D))" .
  }
  note ** = this
  show ?thesis
  proof
    show "?a  rel_interior (convex hull (insert 0 D))"
      unfolding rel_interior_substd_simplex[OF assms(2)] 
    proof safe
      fix i
      assume "i  D"
      have "0 < inverse (2 * real (card D))"
        using d1 by auto
      also have " = ?a  i" using **[of i] i  D
        by auto
      finally show "0 < ?a  i" by auto
    next
      have "sum ((∙) ?a) D = sum (λi. inverse (2 * real (card D))) D"
        by (rule sum.cong) (rule refl, rule **)
      also have " < 1"
        unfolding sum_constant divide_real_def[symmetric]
        by (auto simp add: field_simps)
      finally show "sum ((∙) ?a) D < 1" by auto
    next
      fix i
      assume "i  Basis" and "i  D"
      have "?a  span D"
      proof (rule span_sum[of D "(λb. b /R (2 * real (card D)))" D])
        {
          fix x :: "'a::euclidean_space"
          assume "x  D"
          then have "x  span D"
            using span_base[of _ "D"] by auto
          then have "x /R (2 * real (card D))  span D"
            using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
        }
        then show "x. xD  x /R (2 * real (card D))  span D"
          by auto
      qed
      then show "?a  i = 0 "
        using i  D unfolding span_substd_basis[OF assms(2)] using i  Basis by auto
    qed
  qed
qed

subsectiontag unimportant› ‹Relative interior of convex set›

lemma rel_interior_convex_nonempty_aux:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "0  S"
  shows "rel_interior S  {}"
proof (cases "S = {0}")
  case True
  then show ?thesis using rel_interior_sing by auto
next
  case False
  obtain B where B: "independent B  B  S  S  span B  card B = dim S"
    using basis_exists[of S] by metis
  then have "B  {}"
    using B assms S  {0} span_empty by auto
  have "insert 0 B  span B"
    using subspace_span[of B] subspace_0[of "span B"]
      span_superset by auto
  then have "span (insert 0 B)  span B"
    using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
  then have "convex hull insert 0 B  span B"
    using convex_hull_subset_span[of "insert 0 B"] by auto
  then have "span (convex hull insert 0 B)  span B"
    using span_span[of B]
      span_mono[of "convex hull insert 0 B" "span B"] by blast
  then have *: "span (convex hull insert 0 B) = span B"
    using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  then have "span (convex hull insert 0 B) = span S"
    using B span_mono[of B S] span_mono[of S "span B"]
      span_span[of B] by auto
  moreover have "0  affine hull (convex hull insert 0 B)"
    using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
    using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
      assms hull_subset[of S]
    by auto
  obtain d and f :: "'n  'n" where
    fd: "card d = card B" "linear f" "f ` B = d"
      "f ` span B = {x. iBasis. i  d  x  i = (0::real)}  inj_on f (span B)"
    and d: "d  Basis"
    using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
  then have "bounded_linear f"
    using linear_conv_bounded_linear by auto
  have "d  {}"
    using fd B B  {} by auto
  have "insert 0 d = f ` (insert 0 B)"
    using fd linear_0 by auto
  then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
    using convex_hull_linear_image[of f "(insert 0 d)"]
      convex_hull_linear_image[of f "(insert 0 B)"] linear f
    by auto
  moreover have "rel_interior (f ` (convex hull insert 0 B)) = f ` rel_interior (convex hull insert 0 B)"
  proof (rule rel_interior_injective_on_span_linear_image[OF bounded_linear f])
    show "inj_on f (span (convex hull insert 0 B))"
      using fd * by auto
  qed
  ultimately have "rel_interior (convex hull insert 0 B)  {}"
    using rel_interior_substd_simplex_nonempty[OF d  {} d] by fastforce
  moreover have "convex hull (insert 0 B)  S"
    using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
  ultimately show ?thesis
    using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
qed

lemma rel_interior_eq_empty:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_interior S = {}  S = {}"
proof -
  {
    assume "S  {}"
    then obtain a where "a  S" by auto
    then have "0  (+) (-a) ` S"
      using assms exI[of "(λx. x  S  - a + x = 0)" a] by auto
    then have "rel_interior ((+) (-a) ` S)  {}"
      using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
        convex_translation[of S "-a"] assms
      by auto
    then have "rel_interior S  {}"
      using rel_interior_translation [of "- a"] by simp
  }
  then show ?thesis by auto
qed

lemma interior_simplex_nonempty:
  fixes S :: "'N :: euclidean_space set"
  assumes "independent S" "finite S" "card S = DIM('N)"
  obtains a where "a  interior (convex hull (insert 0 S))"
proof -
  have "affine hull (insert 0 S) = UNIV"
    by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
         assms(1) assms(3) dim_eq_card_independent)
  moreover have "rel_interior (convex hull insert 0 S)  {}"
    using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
  ultimately have "interior (convex hull insert 0 S)  {}"
    by (simp add: rel_interior_interior)
  with that show ?thesis
    by auto
qed

lemma convex_rel_interior:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "convex (rel_interior S)"
proof -
  {
    fix x y and u :: real
    assume assm: "x  rel_interior S" "y  rel_interior S" "0  u" "u  1"
    then have "x  S"
      using rel_interior_subset by auto
    have "x - u *R (x-y)  rel_interior S"
    proof (cases "0 = u")
      case False
      then have "0 < u" using assm by auto
      then show ?thesis
        using assm rel_interior_convex_shrink[of S y x u] assms x  S by auto
    next
      case True
      then show ?thesis using assm by auto
    qed
    then have "(1 - u) *R x + u *R y  rel_interior S"
      by (simp add: algebra_simps)
  }
  then show ?thesis
    unfolding convex_alt by auto
qed

lemma convex_closure_rel_interior:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "closure (rel_interior S) = closure S"
proof -
  have h1: "closure (rel_interior S)  closure S"
    using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
  show ?thesis
  proof (cases "S = {}")
    case False
    then obtain a where a: "a  rel_interior S"
      using rel_interior_eq_empty assms by auto
    { fix x
      assume x: "x  closure S"
      {
        assume "x = a"
        then have "x  closure (rel_interior S)"
          using a unfolding closure_def by auto
      }
      moreover
      {
        assume "x  a"
         {
           fix e :: real
           assume "e > 0"
           define e1 where "e1 = min 1 (e/norm (x - a))"
           then have e1: "e1 > 0" "e1  1" "e1 * norm (x - a)  e"
             using x  a e > 0 le_divide_eq[of e1 e "norm (x - a)"]
             by simp_all
           then have *: "x - e1 *R (x - a)  rel_interior S"
             using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
             by auto
           have "y. y  rel_interior S  y  x  dist y x  e"
             using "*" x  a e1 by force
        }
        then have "x islimpt rel_interior S"
          unfolding islimpt_approachable_le by auto
        then have "x  closure(rel_interior S)"
          unfolding closure_def by auto
      }
      ultimately have "x  closure(rel_interior S)" by auto
    }
    then show ?thesis using h1 by auto
  qed auto
qed

lemma rel_interior_same_affine_hull:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "affine hull (rel_interior S) = affine hull S"
  by (metis assms closure_same_affine_hull convex_closure_rel_interior)

lemma rel_interior_aff_dim:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "aff_dim (rel_interior S) = aff_dim S"
  by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)

lemma rel_interior_rel_interior:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_interior (rel_interior S) = rel_interior S"
proof -
  have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
    using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
  then show ?thesis
    using rel_interior_def by auto
qed

lemma rel_interior_rel_open:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_open (rel_interior S)"
  unfolding rel_open_def using rel_interior_rel_interior assms by auto

lemma convex_rel_interior_closure_aux:
  fixes x y z :: "'n::euclidean_space"
  assumes "0 < a" "0 < b" "(a + b) *R z = a *R x + b *R y"
  obtains e where "0 < e" "e < 1" "z = y - e *R (y - x)"
proof -
  define e where "e = a / (a + b)"
  have "z = (1 / (a + b)) *R ((a + b) *R z)"
    using assms  by (simp add: eq_vector_fraction_iff)
  also have " = (1 / (a + b)) *R (a *R x + b *R y)"
    using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *R z" "a *R x + b *R y"]
    by auto
  also have " = y - e *R (y-x)"
    using e_def assms
    by (simp add: divide_simps vector_fraction_eq_iff) (simp add: algebra_simps)
  finally have "z = y - e *R (y-x)"
    by auto
  moreover have "e > 0" "e < 1" using e_def assms by auto
  ultimately show ?thesis using that[of e] by auto
qed

lemma convex_rel_interior_closure:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_interior (closure S) = rel_interior S"
proof (cases "S = {}")
  case True
  then show ?thesis
    using assms rel_interior_eq_empty by auto
next
  case False
  have "rel_interior (closure S)  rel_interior S"
    using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
    by auto
  moreover
  {
    fix z
    assume z: "z  rel_interior (closure S)"
    obtain x where x: "x  rel_interior S"
      using S  {} assms rel_interior_eq_empty by auto
    have "z  rel_interior S"
    proof (cases "x = z")
      case True
      then show ?thesis using x by auto
    next
      case False
      obtain e where e: "e > 0" "cball z e  affine hull closure S  closure S"
        using z rel_interior_cball[of "closure S"] by auto
      hence *: "0 < e/norm(z-x)" using e False by auto
      define y where "y = z + (e/norm(z-x)) *R (z-x)"
      have yball: "y  cball z e"
        using y_def dist_norm[of z y] e by auto
      have "x  affine hull closure S"
        using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
      moreover have "z  affine hull closure S"
        using z rel_interior_subset hull_subset[of "closure S"] by blast
      ultimately have "y  affine hull closure S"
        using y_def affine_affine_hull[of "closure S"]
          mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
      then have "y  closure S" using e yball by auto
      have "(1 + (e/norm(z-x))) *R z = (e/norm(z-x)) *R x + y"
        using y_def by (simp add: algebra_simps)
      then obtain e1 where "0 < e1" "e1 < 1" "z = y - e1 *R (y - x)"
        using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
        by (auto simp add: algebra_simps)
      then show ?thesis
        using rel_interior_closure_convex_shrink assms x y  closure S
        by fastforce
    qed
  }
  ultimately show ?thesis by auto
qed

lemma convex_interior_closure:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "interior (closure S) = interior S"
  using closure_aff_dim[of S] interior_rel_interior_gen[of S]
    interior_rel_interior_gen[of "closure S"]
    convex_rel_interior_closure[of S] assms
  by auto

lemma open_subset_closure_of_interval:
  assumes "open U" "is_interval S"
  shows "U  closure S  U  interior S"
  by (metis assms convex_interior_closure is_interval_convex open_subset_interior)

lemma closure_eq_rel_interior_eq:
  fixes S1 S2 :: "'n::euclidean_space set"
  assumes "convex S1"
    and "convex S2"
  shows "closure S1 = closure S2  rel_interior S1 = rel_interior S2"
  by (metis convex_rel_interior_closure convex_closure_rel_interior assms)

lemma closure_eq_between:
  fixes S1 S2 :: "'n::euclidean_space set"
  assumes "convex S1"
    and "convex S2"
  shows "closure S1 = closure S2  rel_interior S1  S2  S2  closure S1"
  (is "?A  ?B")
proof
  assume ?A
  then show ?B
    by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
next
  assume ?B
  then have "closure S1  closure S2"
    by (metis assms(1) convex_closure_rel_interior closure_mono)
  moreover from ?B have "closure S1  closure S2"
    by (metis closed_closure closure_minimal)
  ultimately show ?A ..
qed

lemma open_inter_closure_rel_interior:
  fixes S A :: "'n::euclidean_space set"
  assumes "convex S"
    and "open A"
  shows "A  closure S = {}  A  rel_interior S = {}"
  by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)

lemma rel_interior_open_segment:
  fixes a :: "'a :: euclidean_space"
  shows "rel_interior(open_segment a b) = open_segment a b"
proof (cases "a = b")
  case True then show ?thesis by auto
next
  case False then
  have "open_segment a b = affine hull {a, b}  ball ((a + b) /R 2) (norm (b - a) / 2)"
    by (simp add: open_segment_as_ball)
  then show ?thesis
    unfolding rel_interior_eq openin_open
    by (metis Elementary_Metric_Spaces.open_ball False affine_hull_open_segment)
qed

lemma rel_interior_closed_segment:
  fixes a :: "'a :: euclidean_space"
  shows "rel_interior(closed_segment a b) =
         (if a = b then {a} else open_segment a b)"
proof (cases "a = b")
  case True then show ?thesis by auto
next
  case False then show ?thesis
    by simp
       (metis closure_open_segment convex_open_segment convex_rel_interior_closure
              rel_interior_open_segment)
qed

lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment

subsection‹The relative frontier of a set›

definitiontag important› "rel_frontier S = closure S - rel_interior S"

lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
  by (simp add: rel_frontier_def)

lemma rel_frontier_eq_empty:
    fixes S :: "'n::euclidean_space set"
    shows "rel_frontier S = {}  affine S"
  unfolding rel_frontier_def
  using rel_interior_subset_closure  by (auto simp add: rel_interior_eq_closure [symmetric])

lemma rel_frontier_sing [simp]:
    fixes a :: "'n::euclidean_space"
    shows "rel_frontier {a} = {}"
  by (simp add: rel_frontier_def)

lemma rel_frontier_affine_hull:
  fixes S :: "'a::euclidean_space set"
  shows "rel_frontier S  affine hull S"
using closure_affine_hull rel_frontier_def by fastforce

lemma rel_frontier_cball [simp]:
    fixes a :: "'n::euclidean_space"
    shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
proof (cases rule: linorder_cases [of r 0])
  case less then show ?thesis
    by (force simp: sphere_def)
next
  case equal then show ?thesis by simp
next
  case greater then show ?thesis
    by simp (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
qed

lemma rel_frontier_translation:
  fixes a :: "'a::euclidean_space"
  shows "rel_frontier((λx. a + x) ` S) = (λx. a + x) ` (rel_frontier S)"
  by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)

lemma rel_frontier_nonempty_interior:
  fixes S :: "'n::euclidean_space set"
  shows "interior S  {}  rel_frontier S = frontier S"
  by (metis frontier_def interior_rel_interior_gen rel_frontier_def)

lemma rel_frontier_frontier:
  fixes S :: "'n::euclidean_space set"
  shows "affine hull S = UNIV  rel_frontier S = frontier S"
  by (simp add: frontier_def rel_frontier_def rel_interior_interior)

lemma closest_point_in_rel_frontier:
   "closed S; S  {}; x  affine hull S - rel_interior S
    closest_point S x  rel_frontier S"
  by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)

lemma closed_rel_frontier [iff]:
  fixes S :: "'n::euclidean_space set"
  shows "closed (rel_frontier S)"
proof -
  have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
    by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
  show ?thesis
  proof (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
    show "closedin (top_of_set (affine hull S)) (rel_frontier S)"
      by (simp add: "*" rel_frontier_def)
  qed simp
qed

lemma closed_rel_boundary:
  fixes S :: "'n::euclidean_space set"
  shows "closed S  closed(S - rel_interior S)"
  by (metis closed_rel_frontier closure_closed rel_frontier_def)

lemma compact_rel_boundary:
  fixes S :: "'n::euclidean_space set"
  shows "compact S  compact(S - rel_interior S)"
  by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)

lemma bounded_rel_frontier:
  fixes S :: "'n::euclidean_space set"
  shows "bounded S  bounded(rel_frontier S)"
by (simp add: bounded_closure bounded_diff rel_frontier_def)

lemma compact_rel_frontier_bounded:
  fixes S :: "'n::euclidean_space set"
  shows "bounded S  compact(rel_frontier S)"
using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast

lemma compact_rel_frontier:
  fixes S :: "'n::euclidean_space set"
  shows "compact S  compact(rel_frontier S)"
by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)

lemma convex_same_rel_interior_closure:
  fixes S :: "'n::euclidean_space set"
  shows "convex S; convex T
          rel_interior S = rel_interior T  closure S = closure T"
by (simp add: closure_eq_rel_interior_eq)

lemma convex_same_rel_interior_closure_straddle:
  fixes S :: "'n::euclidean_space set"
  shows "convex S; convex T
          rel_interior S = rel_interior T 
             rel_interior S  T  T  closure S"
by (simp add: closure_eq_between convex_same_rel_interior_closure)

lemma convex_rel_frontier_aff_dim:
  fixes S1 S2 :: "'n::euclidean_space set"
  assumes "convex S1"
    and "convex S2"
    and "S2  {}"
    and "S1  rel_frontier S2"
  shows "aff_dim S1 < aff_dim S2"
proof -
  have "S1  closure S2"
    using assms unfolding rel_frontier_def by auto
  then have *: "affine hull S1  affine hull S2"
    using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
  then have "aff_dim S1  aff_dim S2"
    using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
      aff_dim_subset[of "affine hull S1" "affine hull S2"]
    by auto
  moreover
  {
    assume eq: "aff_dim S1 = aff_dim S2"
    then have "S1  {}"
      using aff_dim_empty[of S1] aff_dim_empty[of S2] S2  {} by auto
    have **: "affine hull S1 = affine hull S2"
      by (simp_all add: * eq S1  {} affine_dim_equal)
    obtain a where a: "a  rel_interior S1"
      using S1  {} rel_interior_eq_empty assms by auto
    obtain T where T: "open T" "a  T  S1" "T  affine hull S1  S1"
       using mem_rel_interior[of a S1] a by auto
    then have "a  T  closure S2"
      using a assms unfolding rel_frontier_def by auto
    then obtain b where b: "b  T  rel_interior S2"
      using open_inter_closure_rel_interior[of S2 T] assms T by auto
    then have "b  affine hull S1"
      using rel_interior_subset hull_subset[of S2] ** by auto
    then have "b  S1"
      using T b by auto
    then have False
      using b assms unfolding rel_frontier_def by auto
  }
  ultimately show ?thesis
    using less_le by auto
qed

lemma convex_rel_interior_if:
  fixes S ::  "'n::euclidean_space set"
  assumes "convex S"
    and "z  rel_interior S"
  shows "xaffine hull S. m. m > 1  (e. e > 1  e  m  (1 - e) *R x + e *R z  S)"
proof -
  obtain e1 where e1: "e1 > 0  cball z e1  affine hull S  S"
    using mem_rel_interior_cball[of z S] assms by auto
  {
    fix x
    assume x: "x  affine hull S"
    {
      assume "x  z"
      define m where "m = 1 + e1/norm(x-z)"
      hence "m > 1" using e1 x  z by auto
      {
        fix e
        assume e: "e > 1  e  m"
        have "z  affine hull S"
          using assms rel_interior_subset hull_subset[of S] by auto
        then have *: "(1 - e)*R x + e *R z  affine hull S"
          using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
          by auto
        have "norm (z + e *R x - (x + e *R z)) = norm ((e - 1) *R (x - z))"
          by (simp add: algebra_simps)
        also have " = (e - 1) * norm (x-z)"
          using norm_scaleR e by auto
        also have "  (m - 1) * norm (x - z)"
          using e mult_right_mono[of _ _ "norm(x-z)"] by auto
        also have " = (e1 / norm (x - z)) * norm (x - z)"
          using m_def by auto
        also have " = e1"
          using x  z e1 by simp
        finally have **: "norm (z + e *R x - (x + e *R z))  e1"
          by auto
        have "(1 - e)*R x+ e *R z  cball z e1"
          using m_def **
          unfolding cball_def dist_norm
          by (auto simp add: algebra_simps)
        then have "(1 - e) *R x+ e *R z  S"
          using e * e1 by auto
      }
      then have "m. m > 1  (e. e > 1  e  m  (1 - e) *R x + e *R z  S )"
        using m> 1 by auto
    }
    moreover
    {
      assume "x = z"
      define m where "m = 1 + e1"
      then have "m > 1"
        using e1 by auto
      {
        fix e
        assume e: "e > 1  e  m"
        then have "(1 - e) *R x + e *R z  S"
          using e1 x x = z by (auto simp add: algebra_simps)
        then have "(1 - e) *R x + e *R z  S"
          using e by auto
      }
      then have "m. m > 1  (e. e > 1  e  m  (1 - e) *R x + e *R z  S)"
        using m > 1 by auto
    }
    ultimately have "m. m > 1  (e. e > 1  e  m  (1 - e) *R x + e *R z  S )"
      by blast
  }
  then show ?thesis by auto
qed

lemma convex_rel_interior_if2:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  assumes "z  rel_interior S"
  shows "xaffine hull S. e. e > 1  (1 - e)*R x + e *R z  S"
  using convex_rel_interior_if[of S z] assms by auto

lemma convex_rel_interior_only_if:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "S  {}"
  assumes "xS. e. e > 1  (1 - e) *R x + e *R z  S"
  shows "z  rel_interior S"
proof -
  obtain x where x: "x  rel_interior S"
    using rel_interior_eq_empty assms by auto
  then have "x  S"
    using rel_interior_subset by auto
  then obtain e where e: "e > 1  (1 - e) *R x + e *R z  S"
    using assms by auto
  define y where [abs_def]: "y = (1 - e) *R x + e *R z"
  then have "y  S" using e by auto
  define e1 where "e1 = 1/e"
  then have "0 < e1  e1 < 1" using e by auto
  then have "z  =y - (1 - e1) *R (y - x)"
    using e1_def y_def by (auto simp add: algebra_simps)
  then show ?thesis
    using rel_interior_convex_shrink[of S x y "1-e1"] 0 < e1  e1 < 1 y  S x assms
    by auto
qed

lemma convex_rel_interior_iff:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "S  {}"
  shows "z  rel_interior S  (xS. e. e > 1  (1 - e) *R x + e *R z  S)"
  using assms hull_subset[of S "affine"]
    convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
  by auto

lemma convex_rel_interior_iff2:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "S  {}"
  shows "z  rel_interior S  (xaffine hull S. e. e > 1  (1 - e) *R x + e *R z  S)"
  using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
  by auto

lemma convex_interior_iff:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "z  interior S  (x. e. e > 0  z + e *R x  S)"
proof (cases "aff_dim S = int DIM('n)")
  case False
  { assume "z  interior S"
    then have False
      using False interior_rel_interior_gen[of S] by auto }
  moreover
  { assume r: "x. e. e > 0  z + e *R x  S"
    { fix x
      obtain e1 where e1: "e1 > 0  z + e1 *R (x - z)  S"
        using r by auto
      obtain e2 where e2: "e2 > 0  z + e2 *R (z - x)  S"
        using r by auto
      define x1 where [abs_def]: "x1 = z + e1 *R (x - z)"
      then have x1: "x1  affine hull S"
        using e1 hull_subset[of S] by auto
      define x2 where [abs_def]: "x2 = z + e2 *R (z - x)"
      then have x2: "x2  affine hull S"
        using e2 hull_subset[of S] by auto
      have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
        using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
      then have "z = (e2/(e1+e2)) *R x1 + (e1/(e1+e2)) *R x2"
        by (simp add: x1_def x2_def algebra_simps) (simp add: "*" pth_8)
      then have z: "z  affine hull S"
        using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
          x1 x2 affine_affine_hull[of S] *
        by auto
      have "x1 - x2 = (e1 + e2) *R (x - z)"
        using x1_def x2_def by (auto simp add: algebra_simps)
      then have "x = z+(1/(e1+e2)) *R (x1-x2)"
        using e1 e2 by simp
      then have "x  affine hull S"
        using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
          x1 x2 z affine_affine_hull[of S]
        by auto
    }
    then have "affine hull S = UNIV"
      by auto
    then have "aff_dim S = int DIM('n)"
      using aff_dim_affine_hull[of S] by (simp)
    then have False
      using False by auto
  }
  ultimately show ?thesis by auto
next
  case True
  then have "S  {}"
    using aff_dim_empty[of S] by auto
  have *: "affine hull S = UNIV"
    using True affine_hull_UNIV by auto
  {
    assume "z  interior S"
    then have "z  rel_interior S"
      using True interior_rel_interior_gen[of S] by auto
    then have **: "x. e. e > 1  (1 - e) *R x + e *R z  S"
      using convex_rel_interior_iff2[of S z] assms S  {} * by auto
    fix x
    obtain e1 where e1: "e1 > 1" "(1 - e1) *R (z - x) + e1 *R z  S"
      using **[rule_format, of "z-x"] by auto
    define e where [abs_def]: "e = e1 - 1"
    then have "(1 - e1) *R (z - x) + e1 *R z = z + e *R x"
      by (simp add: algebra_simps)
    then have "e > 0" "z + e *R x  S"
      using e1 e_def by auto
    then have "e. e > 0  z + e *R x  S"
      by auto
  }
  moreover
  {
    assume r: "x. e. e > 0  z + e *R x  S"
    {
      fix x
      obtain e1 where e1: "e1 > 0" "z + e1 *R (z - x)  S"
        using r[rule_format, of "z-x"] by auto
      define e where "e = e1 + 1"
      then have "z + e1 *R (z - x) = (1 - e) *R x + e *R z"
        by (simp add: algebra_simps)
      then have "e > 1" "(1 - e)*R x + e *R z  S"
        using e1 e_def by auto
      then have "e. e > 1  (1 - e) *R x + e *R z  S" by auto
    }
    then have "z  rel_interior S"
      using convex_rel_interior_iff2[of S z] assms S  {} by auto
    then have "z  interior S"
      using True interior_rel_interior_gen[of S] by auto
  }
  ultimately show ?thesis by auto
qed


subsubsectiontag unimportant› ‹Relative interior and closure under common operations›

lemma rel_interior_inter_aux: "{rel_interior S |S. S  I}  I"
proof -
  {
    fix y
    assume "y  {rel_interior S |S. S  I}"
    then have y: "S  I. y  rel_interior S"
      by auto
    {
      fix S
      assume "S  I"
      then have "y  S"
        using rel_interior_subset y by auto
    }
    then have "y  I" by auto
  }
  then show ?thesis by auto
qed

lemma convex_closure_rel_interior_inter:
  assumes "SI. convex (S :: 'n::euclidean_space set)"
    and "{rel_interior S |S. S  I}  {}"
  shows "{closure S |S. S  I}  closure ({rel_interior S |S. S  I})"
proof -
  obtain x where x: "SI. x  rel_interior S"
    using assms by auto
  {
    fix y
    assume "y  {closure S |S. S  I}"
    then have y: "S  I. y  closure S"
      by auto
    {
      assume "y = x"
      then have "y  closure ({rel_interior S |S. S  I})"
        using x closure_subset[of "{rel_interior S |S. S  I}"] by auto
    }
    moreover
    {
      assume "y  x"
      { fix e :: real
        assume e: "e > 0"
        define e1 where "e1 = min 1 (e/norm (y - x))"
        then have e1: "e1 > 0" "e1  1" "e1 * norm (y - x)  e"