Theorems · Theorem · convex and discrete geometry

quasiconcaveOn_iff_min_le

∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
  [inst_3 : LinearOrder β] [inst_4 : SMul 𝕜 E] {s : Set E} {f : E → β},
  QuasiconcaveOn 𝕜 s f ↔
    Convex 𝕜 s ∧
      ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → min (f x) (f y) ≤ f (a • x + b • y)

Defined in

Mathlib.Analysis.Convex.Quasiconvex

Cited by

1 results in Mathlib

Foundations

Depth 18 from the axioms · uses propext

Assumes

SemiringPartialOrderAddCommMonoidLinearOrderSMul