Theorems · Theorem · real analysis

ConvexOn.exists_ge_of_centerMass

∀ {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 : LinearOrder β]
  [IsOrderedAddMonoid β] [inst_7 : Module 𝕜 E] [inst_8 : Module 𝕜 β] [IsStrictOrderedModule 𝕜 β] {s : Set E} {f : E → β}
  {w : ι → 𝕜} {p : ι → E} {t : Finset ι},
  ConvexOn 𝕜 s f → (∀ i ∈ t, 0 ≤ w i) → 0 < ∑ i ∈ t, w i → (∀ i ∈ t, p i ∈ s) → ∃ i ∈ t, f (t.centerMass w p) ≤ f (p i)

If a function f is convex on s, then the value it takes at some center of mass of points of s is less than the value it takes on one of those points.

Defined in

Mathlib.Analysis.Convex.Jensen

Cited by

2 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupAddCommGroupLinearOrderIsOrderedAddMonoidModuleModuleIsStrictOrderedModule