concaveOn_iff_convex_hypograph — Mathlib

English

ConcaveOn 𝕜 s f is equivalent to convexity of the hypograph of f over s.

Русский

ConcaveOn 𝕜 s f эквивалентна выпуклости гипогипографа f над s.

LaTeX

$$$ConcaveOn\ 𝕜\ s\ f \iff Convex\ 𝕜\ {p : E \times β | p.1 \in s ∧ p.2 \le f(p.1)}$$$

Lean4

theorem concaveOn_iff_convex_hypograph : ConcaveOn 𝕜 s f ↔ Convex 𝕜 {p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
  convexOn_iff_convex_epigraph (β := βᵒᵈ)

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