Theorems · Theorem · convex and discrete geometry

convex_iff_segment_subset

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

Defined in

Mathlib.Analysis.Convex.Basic

Cited by

6 results in Mathlib

Foundations

Depth 14 from the axioms · uses no axioms

Assumes

SemiringPartialOrderAddCommMonoidSMul