mathlib documentation

analysis.convex.partition_of_unity

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 β†’ 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 β†’ 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

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theorem partition_of_unity.finsum_smul_mem_convex {ΞΉ : Type u_1} {X : Type u_2} {E : Type u_3} [topological_space X] [add_comm_group E] [module ℝ E] {s : set X} (f : partition_of_unity ΞΉ X s) {g : ΞΉ β†’ X β†’ E} {t : set E} {x : X} (hx : x ∈ s) (hg : βˆ€ (i : ΞΉ), ⇑(⇑f i) x β‰  0 β†’ g i x ∈ t) (ht : convex ℝ t) :
finsum (Ξ» (i : ΞΉ), ⇑(⇑f i) x β€’ g i x) ∈ t
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theorem exists_continuous_forall_mem_convex_of_local {X : Type u_2} {E : Type u_3} [topological_space X] [add_comm_group E] [module ℝ E] [normal_space X] [paracompact_space X] [topological_space E] [has_continuous_add E] [has_continuous_smul ℝ E] {t : X β†’ set E} (ht : βˆ€ (x : X), convex ℝ (t x)) (H : βˆ€ (x : X), βˆƒ (U : set X) (H : U ∈ nhds x) (g : X β†’ E), continuous_on g U ∧ βˆ€ (y : X), y ∈ U β†’ g y ∈ t y) :
βˆƒ (g : C(X, E)), βˆ€ (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 β†’ 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 β†’ 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.

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theorem exists_continuous_forall_mem_convex_of_local_const {X : Type u_2} {E : Type u_3} [topological_space X] [add_comm_group E] [module ℝ E] [normal_space X] [paracompact_space X] [topological_space E] [has_continuous_add E] [has_continuous_smul ℝ E] {t : X β†’ set E} (ht : βˆ€ (x : X), convex ℝ (t x)) (H : βˆ€ (x : X), βˆƒ (c : E), βˆ€αΆ  (y : X) in nhds x, c ∈ t y) :
βˆƒ (g : C(X, E)), βˆ€ (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 β†’ 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.