Theorems · Theorem · real analysis

StrictConcaveOn.map_sum_eq_iff

∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_4} {ι : Type u_5} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜]
  [IsStrictOrderedRing 𝕜] [inst_3 : AddCommGroup E] [inst_4 : AddCommGroup β] [inst_5 : PartialOrder β]
  [IsOrderedAddMonoid β] [inst_7 : Module 𝕜 E] [inst_8 : Module 𝕜 β] [IsStrictOrderedModule 𝕜 β] {s : Set E} {f : E → β}
  {t : Finset ι} {w : ι → 𝕜} {p : ι → E},
  StrictConcaveOn 𝕜 s f →
    (∀ i ∈ t, 0 < w i) →
      ∑ i ∈ t, w i = 1 →
        (∀ i ∈ t, p i ∈ s) → (f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) ↔ ∀ j ∈ t, p j = ∑ i ∈ t, w i • p i)

Canonical form of the equality case of Jensen's equality. For a strictly concave function f and positive weights w, we have f (∑ i ∈ t, w i • p i) = ∑ i ∈ t, w i • f (p i) if and only if the points p are all equal (and in fact all equal to their center of mass w.r.t. w).

Defined in

Mathlib.Analysis.Convex.Jensen

Cited by

0 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupAddCommGroupPartialOrderIsOrderedAddMonoidModuleModuleIsStrictOrderedModule