convexoconvexa

Convexoconvexa is a term that appears in some discussions of convex analysis to denote a function that is simultaneously convex and concave on a given domain. Because a function that is both convex and concave on a convex domain must be affine, the concept is often used to highlight when a function has two-sided convexity properties that force linearity. The term is not universally standardized, but it is occasionally encountered in expository or regional usage.

Definition and basic fact: Let D be a convex subset of a real vector space and f: D → R. The function f is convexoconvexa if for all x, y in D and t in [0,1], f(tx + (1−t)y) ≤ t f(x) + (1−t) f(y) and f(tx + (1−t)y) ≥ t f(x) + (1−t) f(y). From these two inequalities it follows that equality holds for all x, y, t, so f is affine. Consequently, convexoconvexa functions are precisely the affine functions on D, i.e., f(x) = a·x + b for some vector a and scalar b.

Properties and implications: Since affine functions are continuous and differentiable everywhere on D, convexoconvexa functions have a constant gradient, and their behavior is fully determined by a linear part and a constant offset. In optimization, a convexoconvexa objective reduces problems to linear or affine considerations, often simplifying both theory and computation.

Examples and non-examples: In R^n, any function of the form f(x) = a·x + b is convexoconvexa. Constant functions and linear functions are typical examples. A function like f(x) = x^2 is convex but not concave, hence not convexoconvexa.

See also: convex function, concave function, affine function, duality in convex analysis.