Theorems · Theorem · convex and discrete geometry

convexOn_iff_pairwise_pos

∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
  [inst_3 : AddCommMonoid β] [inst_4 : PartialOrder β] [inst_5 : Module 𝕜 E] [inst_6 : Module 𝕜 β] {s : Set E}
  {f : E → β},
  ConvexOn 𝕜 s f ↔
    Convex 𝕜 s ∧ s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y

Defined in

Mathlib.Analysis.Convex.Function

Cited by

3 results in Mathlib

Foundations

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

Assumes

SemiringPartialOrderAddCommMonoidAddCommMonoidPartialOrderModuleModule