Theorems · Theorem · convex and discrete geometry

convex_iff_forall_pos

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

Defined in

Mathlib.Analysis.Convex.Basic

Cited by

4 results in Mathlib

Foundations

Depth 16 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderAddCommMonoidModule