Theorems · Theorem · convex and discrete geometry

ConvexOn.lt_right_of_left_lt

∀ {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] [IsOrderedCancelAddMonoid β] [inst_6 : Module 𝕜 E]
  [inst_7 : Module 𝕜 β] [PosSMulStrictMono 𝕜 β] {s : Set E} {f : E → β},
  ConvexOn 𝕜 s f → ∀ {x y z : E}, x ∈ s → y ∈ s → z ∈ openSegment 𝕜 x y → f x < f z → f z < f y

Defined in

Mathlib.Analysis.Convex.Function

Cited by

2 results in Mathlib

Foundations

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

Assumes

SemiringPartialOrderAddCommMonoidAddCommMonoidLinearOrderIsOrderedCancelAddMonoidModuleModulePosSMulStrictMono