Theorems · Theorem · real analysis

ConvexOn.le_max_of_mem_Icc

∀ {𝕜 : Type u_1} {β : Type u_4} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [inst_3 : AddCommGroup β] [inst_4 : LinearOrder β] [IsOrderedAddMonoid β] [inst_6 : Module 𝕜 β]
  [IsStrictOrderedModule 𝕜 β] {s : Set 𝕜} {f : 𝕜 → β} {x y z : 𝕜},
  ConvexOn 𝕜 s f → x ∈ s → y ∈ s → z ∈ Set.Icc x y → f z ≤ max (f x) (f y)

Maximum principle for convex functions on an interval. If a function f is convex on the interval [x, y], then the eventual maximum of f on [x, y] is at x or y.

Defined in

Mathlib.Analysis.Convex.Jensen

Cited by

1 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupLinearOrderIsOrderedAddMonoidModuleIsStrictOrderedModule