Theorems · Theorem · convex and discrete geometry

StrictConcaveOn.lt_on_openSegment

∀ {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 : LinearOrder β] [IsOrderedAddMonoid β] [inst_6 : SMul 𝕜 E] [inst_7 : Module 𝕜 β]
  [PosSMulStrictMono 𝕜 β] {s : Set E} {f : E → β},
  StrictConcaveOn 𝕜 s f → ∀ {x y z : E}, x ∈ s → y ∈ s → x ≠ y → z ∈ openSegment 𝕜 x y → min (f x) (f y) < f z

A strictly concave function on an open segment is strictly lower-bounded by the min of its endpoints.

Defined in

Mathlib.Analysis.Convex.Function

Cited by

0 results in Mathlib

Foundations

Depth 20 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderAddCommMonoidAddCommMonoidLinearOrderIsOrderedAddMonoidSMulModulePosSMulStrictMono