Partition of unity and convex sets

In this file we prove the following lemma, see exists_continuous_forall_mem_convex_of_local. Let X be a normal paracompact topological space (e.g., any extended metric space). Let E be a topological real vector space. Let t : X → Set E be a family of convex sets. Suppose that for each point x : X, there exists a neighborhood U ∈ 𝓝 X and a function g : X → E that is continuous on U and sends each y ∈ U to a point of t y. Then there exists a continuous map g : C(X, E) such that g x ∈ t x for all x.

We also formulate a useful corollary, see exists_continuous_forall_mem_convex_of_local_const, that assumes that local functions g are constants.

Tags

partition of unity

theorem PartitionOfUnity.finsum_smul_mem_convex {ι : Type u_1} {X : Type u_2} {E : Type u_3} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E] {s : Set X} (f : PartitionOfUnity ι X s) {g : ι → X → E} {t : Set E} {x : X} (hx : x ∈ s) (hg : ∀ (i : ι), (f i) x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t) : (finsum fun (i : ι) => (f i) x • g i x) ∈ t

theorem exists_continuous_forall_mem_convex_of_local {X : Type u_2} {E : Type u_3} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E] [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {t : X → Set E} (ht : ∀ (x : X), Convex ℝ (t x)) (H : ∀ (x : X), ∃ U ∈ nhds x, ∃ (g : X → E), ContinuousOn g U ∧ ∀ y ∈ U, g y ∈ t y) : ∃ (g : C(X, E)), ∀ (x : X), g x ∈ t x

Let X be a normal paracompact topological space (e.g., any extended metric space). Let E be a topological real vector space. Let t : X → Set E be a family of convex sets. Suppose that for each point x : X, there exists a neighborhood U ∈ 𝓝 X and a function g : X → E that is continuous on U and sends each y ∈ U to a point of t y. Then there exists a continuous map g : C(X, E) such that g x ∈ t x for all x. See also exists_continuous_forall_mem_convex_of_local_const.

theorem exists_continuous_forall_mem_convex_of_local_const {X : Type u_2} {E : Type u_3} [TopologicalSpace X] [AddCommGroup E] [Module ℝ E] [NormalSpace X] [ParacompactSpace X] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {t : X → Set E} (ht : ∀ (x : X), Convex ℝ (t x)) (H : ∀ (x : X), ∃ (c : E), ∀ᶠ (y : X) in nhds x, c ∈ t y) : ∃ (g : C(X, E)), ∀ (x : X), g x ∈ t x

Let X be a normal paracompact topological space (e.g., any extended metric space). Let E be a topological real vector space. Let t : X → Set E be a family of convex sets. Suppose that for each point x : X, there exists a vector c : E that belongs to t y for all y in a neighborhood of x. Then there exists a continuous map g : C(X, E) such that g x ∈ t x for all x. See also exists_continuous_forall_mem_convex_of_local.