Theorems · Theorem · convex and discrete geometry

ConvexOn.openSegment_subset_strict_epigraph

∀ {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] [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 →
    ∀ (p q : E × β), p.1 ∈ s ∧ f p.1 < p.2 → q.1 ∈ s ∧ f q.1 ≤ q.2 → openSegment 𝕜 p q ⊆ {p | p.1 ∈ s ∧ f p.1 < p.2}

Defined in

Mathlib.Analysis.Convex.Function

Cited by

2 results in Mathlib

Foundations

Depth 18 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderAddCommMonoidAddCommMonoidPartialOrderIsOrderedCancelAddMonoidModuleModulePosSMulStrictMono