le_sup_of_mem_convexHull — Mathlib

English

For a centered convex combination, equality in Jensen's inequality implies a relation among the involved points.

Русский

При центрированных выпуклых сочетаниях равенство Дженсона дает зависимость между точками.

LaTeX

$$$ \text{ConvexOn}_{\mathbb{K}}(s,f) \Rightarrow f( t\centerMass(w,p) ) \le t\centerMass(w, f\circ p) $$$

Lean4

theorem le_sup_of_mem_convexHull {t : Finset E} (hf : ConvexOn 𝕜 s f) (hts : ↑t ⊆ s)
    (hx : x ∈ convexHull 𝕜 (t : Set E)) : f x ≤ t.sup' (coe_nonempty.1 <| convexHull_nonempty_iff.1 ⟨x, hx⟩) f :=
  by
  obtain ⟨w, hw₀, hw₁, rfl⟩ := mem_convexHull.1 hx
  exact (hf.map_centerMass_le hw₀ (by positivity) hts).trans (centerMass_le_sup hw₀ <| by positivity)

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