English
If f is convex on the whole domain, then the inverse f^{-1} is concave on the whole domain.
Русский
Если f выпукла на всей области определения, то её обратная функция выпукла в смысле обратной кривизны, т.е. конкавна.
LaTeX
$$$$\\text{If } f:\\alpha \\to \\beta \\text{ is an order isomorphism and } f \\text{ is convex on } \\mathsf{univ}, \\text{ then } f^{-1}:\\beta \\to \\alpha \\text{ is concave on } \\mathsf{univ}.$$$$
Lean4
theorem concaveOn_symm (f : α ≃o β) (hf : ConvexOn 𝕜 univ f) : ConcaveOn 𝕜 univ f.symm :=
by
refine ⟨convex_univ, fun x _ y _ a b ha hb hab => ?_⟩
obtain ⟨x', hx''⟩ := f.surjective.exists.mp ⟨x, rfl⟩
obtain ⟨y', hy''⟩ := f.surjective.exists.mp ⟨y, rfl⟩
simp only [hx'', hy'', OrderIso.symm_apply_apply]
rw [← f.le_iff_le, OrderIso.apply_symm_apply]
exact hf.2 (by simp : x' ∈ univ) (by simp : y' ∈ univ) ha hb hab
