Documentation

Mathlib.Analysis.Convex.Strict

Strictly convex sets #

This file defines strictly convex sets.

A set is strictly convex if the open segment between any two distinct points lies in its interior.

def StrictConvex (๐•œ : Type u_1) {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] (s : Set E) :

A set is strictly convex if the open segment between any two distinct points lies is in its interior. This basically means "convex and not flat on the boundary".

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Instances For
    theorem strictConvex_iff_openSegment_subset {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] {s : Set E} :
    StrictConvex ๐•œ s โ†” s.Pairwise fun (x y : E) => openSegment ๐•œ x y โŠ† interior s
    theorem StrictConvex.openSegment_subset {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] {s : Set E} {x : E} {y : E} (hs : StrictConvex ๐•œ s) (hx : x โˆˆ s) (hy : y โˆˆ s) (h : x โ‰  y) :
    openSegment ๐•œ x y โŠ† interior s
    theorem strictConvex_empty {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] :
    theorem strictConvex_univ {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] :
    StrictConvex ๐•œ Set.univ
    theorem StrictConvex.eq {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] {s : Set E} {x : E} {y : E} {a : ๐•œ} {b : ๐•œ} (hs : StrictConvex ๐•œ s) (hx : x โˆˆ s) (hy : y โˆˆ s) (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) (h : a โ€ข x + b โ€ข y โˆ‰ interior s) :
    x = y
    theorem StrictConvex.inter {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] {s : Set E} {t : Set E} (hs : StrictConvex ๐•œ s) (ht : StrictConvex ๐•œ t) :
    StrictConvex ๐•œ (s โˆฉ t)
    theorem Directed.strictConvex_iUnion {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] {ฮน : Sort u_6} {s : ฮน โ†’ Set E} (hdir : Directed (fun (x x_1 : Set E) => x โŠ† x_1) s) (hs : โˆ€ โฆƒi : ฮนโฆ„, StrictConvex ๐•œ (s i)) :
    StrictConvex ๐•œ (โ‹ƒ (i : ฮน), s i)
    theorem DirectedOn.strictConvex_sUnion {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [SMul ๐•œ E] {S : Set (Set E)} (hdir : DirectedOn (fun (x x_1 : Set E) => x โŠ† x_1) S) (hS : โˆ€ s โˆˆ S, StrictConvex ๐•œ s) :
    theorem StrictConvex.convex {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [Module ๐•œ E] {s : Set E} (hs : StrictConvex ๐•œ s) :
    Convex ๐•œ s
    theorem Convex.strictConvex_of_isOpen {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [Module ๐•œ E] {s : Set E} (h : IsOpen s) (hs : Convex ๐•œ s) :
    StrictConvex ๐•œ s

    An open convex set is strictly convex.

    theorem IsOpen.strictConvex_iff {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [Module ๐•œ E] {s : Set E} (h : IsOpen s) :
    StrictConvex ๐•œ s โ†” Convex ๐•œ s
    theorem strictConvex_singleton {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [Module ๐•œ E] (c : E) :
    StrictConvex ๐•œ {c}
    theorem Set.Subsingleton.strictConvex {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommMonoid E] [Module ๐•œ E] {s : Set E} (hs : s.Subsingleton) :
    StrictConvex ๐•œ s
    theorem StrictConvex.linear_image {๐•œ : Type u_1} {๐• : Type u_2} {E : Type u_3} {F : Type u_4} [OrderedSemiring ๐•œ] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module ๐•œ E] [Module ๐•œ F] {s : Set E} [Semiring ๐•] [Module ๐• E] [Module ๐• F] [LinearMap.CompatibleSMul E F ๐•œ ๐•] (hs : StrictConvex ๐•œ s) (f : E โ†’โ‚—[๐•] F) (hf : IsOpenMap โ‡‘f) :
    StrictConvex ๐•œ (โ‡‘f '' s)
    theorem StrictConvex.is_linear_image {๐•œ : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedSemiring ๐•œ] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module ๐•œ E] [Module ๐•œ F] {s : Set E} (hs : StrictConvex ๐•œ s) {f : E โ†’ F} (h : IsLinearMap ๐•œ f) (hf : IsOpenMap f) :
    StrictConvex ๐•œ (f '' s)
    theorem StrictConvex.linear_preimage {๐•œ : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedSemiring ๐•œ] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module ๐•œ E] [Module ๐•œ F] {s : Set F} (hs : StrictConvex ๐•œ s) (f : E โ†’โ‚—[๐•œ] F) (hf : Continuous โ‡‘f) (hfinj : Function.Injective โ‡‘f) :
    StrictConvex ๐•œ (โ‡‘f โปยน' s)
    theorem StrictConvex.is_linear_preimage {๐•œ : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedSemiring ๐•œ] [TopologicalSpace E] [TopologicalSpace F] [AddCommMonoid E] [AddCommMonoid F] [Module ๐•œ E] [Module ๐•œ F] {s : Set F} (hs : StrictConvex ๐•œ s) {f : E โ†’ F} (h : IsLinearMap ๐•œ f) (hf : Continuous f) (hfinj : Function.Injective f) :
    theorem Set.OrdConnected.strictConvex {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] {s : Set ฮฒ} (hs : s.OrdConnected) :
    StrictConvex ๐•œ s
    theorem strictConvex_Iic {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) :
    StrictConvex ๐•œ (Set.Iic r)
    theorem strictConvex_Ici {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) :
    StrictConvex ๐•œ (Set.Ici r)
    theorem strictConvex_Iio {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) :
    StrictConvex ๐•œ (Set.Iio r)
    theorem strictConvex_Ioi {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) :
    StrictConvex ๐•œ (Set.Ioi r)
    theorem strictConvex_Icc {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) (s : ฮฒ) :
    StrictConvex ๐•œ (Set.Icc r s)
    theorem strictConvex_Ioo {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) (s : ฮฒ) :
    StrictConvex ๐•œ (Set.Ioo r s)
    theorem strictConvex_Ico {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) (s : ฮฒ) :
    StrictConvex ๐•œ (Set.Ico r s)
    theorem strictConvex_Ioc {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) (s : ฮฒ) :
    StrictConvex ๐•œ (Set.Ioc r s)
    theorem strictConvex_uIcc {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) (s : ฮฒ) :
    StrictConvex ๐•œ (Set.uIcc r s)
    theorem strictConvex_uIoc {๐•œ : Type u_1} {ฮฒ : Type u_5} [OrderedSemiring ๐•œ] [TopologicalSpace ฮฒ] [LinearOrderedCancelAddCommMonoid ฮฒ] [OrderTopology ฮฒ] [Module ๐•œ ฮฒ] [OrderedSMul ๐•œ ฮฒ] (r : ฮฒ) (s : ฮฒ) :
    StrictConvex ๐•œ (ฮ™ r s)
    theorem StrictConvex.preimage_add_right {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCancelCommMonoid E] [ContinuousAdd E] [Module ๐•œ E] {s : Set E} (hs : StrictConvex ๐•œ s) (z : E) :
    StrictConvex ๐•œ ((fun (x : E) => z + x) โปยน' s)

    The translation of a strictly convex set is also strictly convex.

    theorem StrictConvex.preimage_add_left {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCancelCommMonoid E] [ContinuousAdd E] [Module ๐•œ E] {s : Set E} (hs : StrictConvex ๐•œ s) (z : E) :
    StrictConvex ๐•œ ((fun (x : E) => x + z) โปยน' s)

    The translation of a strictly convex set is also strictly convex.

    theorem StrictConvex.add {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] [ContinuousAdd E] {s : Set E} {t : Set E} (hs : StrictConvex ๐•œ s) (ht : StrictConvex ๐•œ t) :
    StrictConvex ๐•œ (s + t)
    theorem StrictConvex.add_left {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] [ContinuousAdd E] {s : Set E} (hs : StrictConvex ๐•œ s) (z : E) :
    StrictConvex ๐•œ ((fun (x : E) => z + x) '' s)
    theorem StrictConvex.add_right {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] [ContinuousAdd E] {s : Set E} (hs : StrictConvex ๐•œ s) (z : E) :
    StrictConvex ๐•œ ((fun (x : E) => x + z) '' s)
    theorem StrictConvex.vadd {๐•œ : Type u_1} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] [ContinuousAdd E] {s : Set E} (hs : StrictConvex ๐•œ s) (x : E) :
    StrictConvex ๐•œ (x +แตฅ s)

    The translation of a strictly convex set is also strictly convex.

    theorem StrictConvex.smul {๐•œ : Type u_1} {๐• : Type u_2} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] [LinearOrderedField ๐•] [Module ๐• E] [ContinuousConstSMul ๐• E] [LinearMap.CompatibleSMul E E ๐•œ ๐•] {s : Set E} (hs : StrictConvex ๐•œ s) (c : ๐•) :
    StrictConvex ๐•œ (c โ€ข s)
    theorem StrictConvex.affinity {๐•œ : Type u_1} {๐• : Type u_2} {E : Type u_3} [OrderedSemiring ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] [LinearOrderedField ๐•] [Module ๐• E] [ContinuousConstSMul ๐• E] [LinearMap.CompatibleSMul E E ๐•œ ๐•] {s : Set E} [ContinuousAdd E] (hs : StrictConvex ๐•œ s) (z : E) (c : ๐•) :
    StrictConvex ๐•œ (z +แตฅ c โ€ข s)
    theorem StrictConvex.preimage_smul {๐•œ : Type u_1} {E : Type u_3} [OrderedCommSemiring ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] [NoZeroSMulDivisors ๐•œ E] [ContinuousConstSMul ๐•œ E] {s : Set E} (hs : StrictConvex ๐•œ s) (c : ๐•œ) :
    StrictConvex ๐•œ ((fun (z : E) => c โ€ข z) โปยน' s)
    theorem StrictConvex.eq_of_openSegment_subset_frontier {๐•œ : Type u_1} {E : Type u_3} [OrderedRing ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} {x : E} {y : E} [Nontrivial ๐•œ] [DenselyOrdered ๐•œ] (hs : StrictConvex ๐•œ s) (hx : x โˆˆ s) (hy : y โˆˆ s) (h : openSegment ๐•œ x y โŠ† frontier s) :
    x = y
    theorem StrictConvex.add_smul_mem {๐•œ : Type u_1} {E : Type u_3} [OrderedRing ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} {x : E} {y : E} (hs : StrictConvex ๐•œ s) (hx : x โˆˆ s) (hxy : x + y โˆˆ s) (hy : y โ‰  0) {t : ๐•œ} (htโ‚€ : 0 < t) (htโ‚ : t < 1) :
    theorem StrictConvex.smul_mem_of_zero_mem {๐•œ : Type u_1} {E : Type u_3} [OrderedRing ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} {x : E} (hs : StrictConvex ๐•œ s) (zero_mem : 0 โˆˆ s) (hx : x โˆˆ s) (hxโ‚€ : x โ‰  0) {t : ๐•œ} (htโ‚€ : 0 < t) (htโ‚ : t < 1) :
    theorem StrictConvex.add_smul_sub_mem {๐•œ : Type u_1} {E : Type u_3} [OrderedRing ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} {x : E} {y : E} (h : StrictConvex ๐•œ s) (hx : x โˆˆ s) (hy : y โˆˆ s) (hxy : x โ‰  y) {t : ๐•œ} (htโ‚€ : 0 < t) (htโ‚ : t < 1) :
    theorem StrictConvex.affine_preimage {๐•œ : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedRing ๐•œ] [TopologicalSpace E] [TopologicalSpace F] [AddCommGroup E] [AddCommGroup F] [Module ๐•œ E] [Module ๐•œ F] {s : Set F} (hs : StrictConvex ๐•œ s) {f : E โ†’แตƒ[๐•œ] F} (hf : Continuous โ‡‘f) (hfinj : Function.Injective โ‡‘f) :
    StrictConvex ๐•œ (โ‡‘f โปยน' s)

    The preimage of a strictly convex set under an affine map is strictly convex.

    theorem StrictConvex.affine_image {๐•œ : Type u_1} {E : Type u_3} {F : Type u_4} [OrderedRing ๐•œ] [TopologicalSpace E] [TopologicalSpace F] [AddCommGroup E] [AddCommGroup F] [Module ๐•œ E] [Module ๐•œ F] {s : Set E} (hs : StrictConvex ๐•œ s) {f : E โ†’แตƒ[๐•œ] F} (hf : IsOpenMap โ‡‘f) :
    StrictConvex ๐•œ (โ‡‘f '' s)

    The image of a strictly convex set under an affine map is strictly convex.

    theorem StrictConvex.neg {๐•œ : Type u_1} {E : Type u_3} [OrderedRing ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} [TopologicalAddGroup E] (hs : StrictConvex ๐•œ s) :
    StrictConvex ๐•œ (-s)
    theorem StrictConvex.sub {๐•œ : Type u_1} {E : Type u_3} [OrderedRing ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} {t : Set E} [TopologicalAddGroup E] (hs : StrictConvex ๐•œ s) (ht : StrictConvex ๐•œ t) :
    StrictConvex ๐•œ (s - t)
    theorem strictConvex_iff_div {๐•œ : Type u_1} {E : Type u_3} [LinearOrderedField ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} :
    StrictConvex ๐•œ s โ†” s.Pairwise fun (x y : E) => โˆ€ โฆƒa b : ๐•œโฆ„, 0 < a โ†’ 0 < b โ†’ (a / (a + b)) โ€ข x + (b / (a + b)) โ€ข y โˆˆ interior s

    Alternative definition of set strict convexity, using division.

    theorem StrictConvex.mem_smul_of_zero_mem {๐•œ : Type u_1} {E : Type u_3} [LinearOrderedField ๐•œ] [TopologicalSpace E] [AddCommGroup E] [Module ๐•œ E] {s : Set E} {x : E} (hs : StrictConvex ๐•œ s) (zero_mem : 0 โˆˆ s) (hx : x โˆˆ s) (hxโ‚€ : x โ‰  0) {t : ๐•œ} (ht : 1 < t) :

    Convex sets in an ordered space #

    Relates Convex and Set.OrdConnected.

    @[simp]
    theorem strictConvex_iff_convex {๐•œ : Type u_1} [LinearOrderedField ๐•œ] [TopologicalSpace ๐•œ] [OrderTopology ๐•œ] {s : Set ๐•œ} :
    StrictConvex ๐•œ s โ†” Convex ๐•œ s

    A set in a linear ordered field is strictly convex if and only if it is convex.

    theorem strictConvex_iff_ordConnected {๐•œ : Type u_1} [LinearOrderedField ๐•œ] [TopologicalSpace ๐•œ] [OrderTopology ๐•œ] {s : Set ๐•œ} :
    StrictConvex ๐•œ s โ†” s.OrdConnected
    theorem StrictConvex.ordConnected {๐•œ : Type u_1} [LinearOrderedField ๐•œ] [TopologicalSpace ๐•œ] [OrderTopology ๐•œ] {s : Set ๐•œ} :
    StrictConvex ๐•œ s โ†’ s.OrdConnected

    Alias of the forward direction of strictConvex_iff_ordConnected.