Theorems · Theorem · convex and discrete geometry

neg_strictConvexOn_iff

∀ {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 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : SMul 𝕜 E] [inst_7 : Module 𝕜 β]
  {s : Set E} {f : E → β}, StrictConvexOn 𝕜 s (-f) ↔ StrictConcaveOn 𝕜 s f

A function -f is strictly convex iff f is strictly concave.

Defined in

Mathlib.Analysis.Convex.Function

Cited by

3 results in Mathlib

Foundations

Depth 17 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderAddCommMonoidAddCommGroupPartialOrderIsOrderedAddMonoidSMulModule