Functions that are both convex and concave

On a convex set s, a function is both convex and concave exactly when it preserves the convex combinations of points of s: f (a • x + b • y) = a • f x + b • f y for x, y ∈ s and a, b ≥ 0 with a + b = 1. This recognizes the two properties from the affine combination identity, for instance for a Lagrangian that is affine in each variable separately, whose convexity and concavity feed Sion's minimax theorem.

theorem Convex.convexOn_and_concaveOn_iff {𝕜 : Type u_1} {E : Type u_2} {β : Type u_3} [Semiring 𝕜] [PartialOrder 𝕜] [AddCommMonoid E] [SMul 𝕜 E] [AddCommMonoid β] [PartialOrder β] [SMul 𝕜 β] {s : Set E} {f : E → β} (hs : Convex 𝕜 s) : ConvexOn 𝕜 s f ∧ ConcaveOn 𝕜 s f ↔ ∀ x ∈ s, ∀ y ∈ s, ∀ (a b : 𝕜), 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) = a • f x + b • f y

On a convex set, a function is both convex and concave if and only if it preserves the convex combinations of points of the set. The identity is only required for the combinations used by ConvexOn and ConcaveOn, so it may fail off s and for coefficients outside [0, 1].