Theorems · Theorem · convex and discrete geometry

quasilinearOn_iff_mem_uIcc

∀ {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},
  QuasilinearOn 𝕜 s f ↔
    Convex 𝕜 s ∧
      ∀ ⦃x : E⦄,
        x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ Set.uIcc (f x) (f y)

Defined in

Mathlib.Analysis.Convex.Quasiconvex

Cited by

0 results in Mathlib

Foundations

Depth 19 from the axioms · uses propext, Quot.sound

Assumes

SemiringPartialOrderAddCommMonoidLinearOrderSMul