mathlib3 documentation

analysis.convex.star

Star-convex sets #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This files defines star-convex sets (aka star domains, star-shaped set, radially convex set).

A set is star-convex at x if every segment from x to a point in the set is contained in the set.

This is the prototypical example of a contractible set in homotopy theory (by scaling every point towards x), but has wider uses.

Note that this has nothing to do with star rings, has_star and co.

Main declarations #

star_convex 𝕜 x s: s is star-convex at x with scalars 𝕜.

Implementation notes #

Instead of saying that a set is star-convex, we say a set is star-convex at a point. This has the advantage of allowing us to talk about convexity as being "everywhere star-convexity" and of making the union of star-convex sets be star-convex.

Incidentally, this choice means we don't need to assume a set is nonempty for it to be star-convex. Concretely, the empty set is star-convex at every point.

TODO #

Balanced sets are star-convex.

The closure of a star-convex set is star-convex.

Star-convex sets are contractible.

A nonempty open star-convex set in ℝ^n is diffeomorphic to the entire space.

def star_convex (𝕜 : Type u_1) {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] (x : E) (s : set E) :

Prop

Star-convexity of sets. s is star-convex at x if every segment from x to a point in s is contained in s.

Equations

star_convex 𝕜 x s = ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s

theorem star_convex_iff_segment_subset {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {s : set E} :

star_convex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → segment 𝕜 x y ⊆ s

theorem star_convex.segment_subset {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {s : set E} (h : star_convex 𝕜 x s) {y : E} (hy : y ∈ s) :

segment 𝕜 x y ⊆ s

theorem star_convex.open_segment_subset {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {s : set E} (h : star_convex 𝕜 x s) {y : E} (hy : y ∈ s) :

open_segment 𝕜 x y ⊆ s

theorem star_convex_iff_pointwise_add_subset {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {s : set E} :

star_convex 𝕜 x s ↔ ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • {x} + b • s ⊆ s

Alternative definition of star-convexity, in terms of pointwise set operations.

theorem star_convex_empty {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] (x : E) :

star_convex 𝕜 x ∅

theorem star_convex_univ {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] (x : E) :

star_convex 𝕜 x set.univ

theorem star_convex.inter {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {s t : set E} (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 x t) :

star_convex 𝕜 x (s ∩ t)

theorem star_convex_sInter {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {S : set (set E)} (h : ∀ (s : set E), s ∈ S → star_convex 𝕜 x s) :

star_convex 𝕜 x (⋂₀ S)

theorem star_convex_Inter {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {ι : Sort u_3} {s : ι → set E} (h : ∀ (i : ι), star_convex 𝕜 x (s i)) :

star_convex 𝕜 x (⋂ (i : ι), s i)

theorem star_convex.union {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {s t : set E} (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 x t) :

star_convex 𝕜 x (s ∪ t)

theorem star_convex_Union {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {ι : Sort u_3} {s : ι → set E} (hs : ∀ (i : ι), star_convex 𝕜 x (s i)) :

star_convex 𝕜 x (⋃ (i : ι), s i)

theorem star_convex_sUnion {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [has_smul 𝕜 E] {x : E} {S : set (set E)} (hS : ∀ (s : set E), s ∈ S → star_convex 𝕜 x s) :

star_convex 𝕜 x (⋃₀ S)

theorem star_convex.prod {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ordered_semiring 𝕜] [add_comm_monoid E] [add_comm_monoid F] [has_smul 𝕜 E] [has_smul 𝕜 F] {x : E} {y : F} {s : set E} {t : set F} (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 y t) :

star_convex 𝕜 (x, y) (s ×ˢ t)

theorem star_convex_pi {𝕜 : Type u_1} [ordered_semiring 𝕜] {ι : Type u_2} {E : ι → Type u_3} [Π (i : ι), add_comm_monoid (E i)] [Π (i : ι), has_smul 𝕜 (E i)] {x : Π (i : ι), E i} {s : set ι} {t : Π (i : ι), set (E i)} (ht : ∀ ⦃i : ι⦄, i ∈ s → star_convex 𝕜 (x i) (t i)) :

star_convex 𝕜 x (s.pi t)

theorem star_convex.mem {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 x s) (h : s.nonempty) :

x ∈ s

theorem star_convex_iff_forall_pos {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hx : x ∈ s) :

star_convex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s

theorem star_convex_iff_forall_ne_pos {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hx : x ∈ s) :

star_convex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s

theorem star_convex_iff_open_segment_subset {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hx : x ∈ s) :

star_convex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → open_segment 𝕜 x y ⊆ s

theorem star_convex_singleton {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] (x : E) :

star_convex 𝕜 x {x}

theorem star_convex.linear_image {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ordered_semiring 𝕜] [add_comm_monoid E] [add_comm_monoid F] [module 𝕜 E] [module 𝕜 F] {x : E} {s : set E} (hs : star_convex 𝕜 x s) (f : E →ₗ[𝕜] F) :

star_convex 𝕜 (⇑f x) (⇑f '' s)

theorem star_convex.is_linear_image {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ordered_semiring 𝕜] [add_comm_monoid E] [add_comm_monoid F] [module 𝕜 E] [module 𝕜 F] {x : E} {s : set E} (hs : star_convex 𝕜 x s) {f : E → F} (hf : is_linear_map 𝕜 f) :

star_convex 𝕜 (f x) (f '' s)

theorem star_convex.linear_preimage {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ordered_semiring 𝕜] [add_comm_monoid E] [add_comm_monoid F] [module 𝕜 E] [module 𝕜 F] {x : E} {s : set F} (f : E →ₗ[𝕜] F) (hs : star_convex 𝕜 (⇑f x) s) :

star_convex 𝕜 x (⇑f ⁻¹' s)

theorem star_convex.is_linear_preimage {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ordered_semiring 𝕜] [add_comm_monoid E] [add_comm_monoid F] [module 𝕜 E] [module 𝕜 F] {x : E} {s : set F} {f : E → F} (hs : star_convex 𝕜 (f x) s) (hf : is_linear_map 𝕜 f) :

star_convex 𝕜 x (f ⁻¹' s)

theorem star_convex.add {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x y : E} {s t : set E} (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 y t) :

star_convex 𝕜 (x + y) (s + t)

theorem star_convex.add_left {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 x s) (z : E) :

star_convex 𝕜 (z + x) ((λ (x : E), z + x) '' s)

theorem star_convex.add_right {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 x s) (z : E) :

star_convex 𝕜 (x + z) ((λ (x : E), x + z) '' s)

theorem star_convex.preimage_add_right {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x z : E} {s : set E} (hs : star_convex 𝕜 (z + x) s) :

star_convex 𝕜 x ((λ (x : E), z + x) ⁻¹' s)

The translation of a star-convex set is also star-convex.

theorem star_convex.preimage_add_left {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x z : E} {s : set E} (hs : star_convex 𝕜 (x + z) s) :

star_convex 𝕜 x ((λ (x : E), x + z) ⁻¹' s)

The translation of a star-convex set is also star-convex.

theorem star_convex.sub' {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [add_comm_group E] [module 𝕜 E] {x y : E} {s : set (E × E)} (hs : star_convex 𝕜 (x, y) s) :

star_convex 𝕜 (x - y) ((λ (x : E × E), x.fst - x.snd) '' s)

theorem star_convex.smul {𝕜 : Type u_1} {E : Type u_2} [ordered_comm_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 x s) (c : 𝕜) :

star_convex 𝕜 (c • x) (c • s)

theorem star_convex.preimage_smul {𝕜 : Type u_1} {E : Type u_2} [ordered_comm_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} {c : 𝕜} (hs : star_convex 𝕜 (c • x) s) :

star_convex 𝕜 x ((λ (z : E), c • z) ⁻¹' s)

theorem star_convex.affinity {𝕜 : Type u_1} {E : Type u_2} [ordered_comm_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 x s) (z : E) (c : 𝕜) :

star_convex 𝕜 (z + c • x) ((λ (x : E), z + c • x) '' s)

theorem star_convex_zero_iff {𝕜 : Type u_1} {E : Type u_2} [ordered_ring 𝕜] [add_comm_monoid E] [smul_with_zero 𝕜 E] {s : set E} :

star_convex 𝕜 0 s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s

theorem star_convex.add_smul_mem {𝕜 : Type u_1} {E : Type u_2} [ordered_ring 𝕜] [add_comm_group E] [module 𝕜 E] {x y : E} {s : set E} (hs : star_convex 𝕜 x s) (hy : x + y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t) (ht₁ : t ≤ 1) :

x + t • y ∈ s

theorem star_convex.smul_mem {𝕜 : Type u_1} {E : Type u_2} [ordered_ring 𝕜] [add_comm_group E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t) (ht₁ : t ≤ 1) :

t • x ∈ s

theorem star_convex.add_smul_sub_mem {𝕜 : Type u_1} {E : Type u_2} [ordered_ring 𝕜] [add_comm_group E] [module 𝕜 E] {x y : E} {s : set E} (hs : star_convex 𝕜 x s) (hy : y ∈ s) {t : 𝕜} (ht₀ : 0 ≤ t) (ht₁ : t ≤ 1) :

x + t • (y - x) ∈ s

theorem star_convex.affine_preimage {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ordered_ring 𝕜] [add_comm_group E] [add_comm_group F] [module 𝕜 E] [module 𝕜 F] {x : E} (f : E →ᵃ[𝕜] F) {s : set F} (hs : star_convex 𝕜 (⇑f x) s) :

star_convex 𝕜 x (⇑f ⁻¹' s)

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

theorem star_convex.affine_image {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [ordered_ring 𝕜] [add_comm_group E] [add_comm_group F] [module 𝕜 E] [module 𝕜 F] {x : E} (f : E →ᵃ[𝕜] F) {s : set E} (hs : star_convex 𝕜 x s) :

star_convex 𝕜 (⇑f x) (⇑f '' s)

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

theorem star_convex.neg {𝕜 : Type u_1} {E : Type u_2} [ordered_ring 𝕜] [add_comm_group E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 x s) :

star_convex 𝕜 (-x) (-s)

theorem star_convex.sub {𝕜 : Type u_1} {E : Type u_2} [ordered_ring 𝕜] [add_comm_group E] [module 𝕜 E] {x y : E} {s t : set E} (hs : star_convex 𝕜 x s) (ht : star_convex 𝕜 y t) :

star_convex 𝕜 (x - y) (s - t)

theorem star_convex_iff_div {𝕜 : Type u_1} {E : Type u_2} [linear_ordered_field 𝕜] [add_comm_group E] [module 𝕜 E] {x : E} {s : set E} :

star_convex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b → (a / (a + b)) • x + (b / (a + b)) • y ∈ s

Alternative definition of star-convexity, using division.

theorem star_convex.mem_smul {𝕜 : Type u_1} {E : Type u_2} [linear_ordered_field 𝕜] [add_comm_group E] [module 𝕜 E] {x : E} {s : set E} (hs : star_convex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht : 1 ≤ t) :

x ∈ t • s

Star-convex sets in an ordered space #

Relates star_convex and set.ord_connected.

theorem set.ord_connected.star_convex {𝕜 : Type u_1} {E : Type u_2} [ordered_semiring 𝕜] [ordered_add_comm_monoid E] [module 𝕜 E] [ordered_smul 𝕜 E] {x : E} {s : set E} (hs : s.ord_connected) (hx : x ∈ s) (h : ∀ (y : E), y ∈ s → x ≤ y ∨ y ≤ x) :

star_convex 𝕜 x s

theorem star_convex_iff_ord_connected {𝕜 : Type u_1} [linear_ordered_field 𝕜] {x : 𝕜} {s : set 𝕜} (hx : x ∈ s) :

star_convex 𝕜 x s ↔ s.ord_connected

theorem star_convex.ord_connected {𝕜 : Type u_1} [linear_ordered_field 𝕜] {x : 𝕜} {s : set 𝕜} (hx : x ∈ s) :

star_convex 𝕜 x s → s.ord_connected

Alias of the forward direction of star_convex_iff_ord_connected.