Theorems · Theorem · convex and discrete geometry

neg_convexOn_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}, ConvexOn 𝕜 s (-f) ↔ ConcaveOn 𝕜 s f

A function -f is convex iff f is concave.

Defined in

Mathlib.Analysis.Convex.Function

Cited by

10 results in Mathlib

Foundations

Depth 16 from the axioms · uses propext, Quot.sound

Assumes

SemiringPartialOrderAddCommMonoidAddCommGroupPartialOrderIsOrderedAddMonoidSMulModule