Theorems · Theorem · convex and discrete geometry

Convex.combo_interior_closure_subset_interior

∀ {𝕜 : Type u_2} {E : Type u_3} [inst : Field 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommGroup E]
  [inst_3 : Module 𝕜 E] [inst_4 : TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E},
  Convex 𝕜 s → ∀ {a b : 𝕜}, 0 < a → 0 ≤ b → a + b = 1 → a • interior s + b • closure s ⊆ interior s

If s is a convex set, then a • interior s + b • closure s ⊆ interior s for all 0 < a, 0 ≤ b, a + b = 1. See also Convex.combo_interior_self_subset_interior for a weaker version.

Defined in

Mathlib.Analysis.Convex.Topology

Cited by

3 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldPartialOrderAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousConstSMul