Theorems · Theorem · convex and discrete geometry

convex_iff_openSegment_subset

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
  [inst_3 : Module 𝕜 E] {s : Set E} [ZeroLEOneClass 𝕜],
  Convex 𝕜 s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → openSegment 𝕜 x y ⊆ s

Defined in

Mathlib.Analysis.Convex.Basic

Cited by

4 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderAddCommMonoidModuleZeroLEOneClass