convexities

Convexities encompass a family of notions in mathematics that express a stability of structure under averaging and linear interpolation. The core ideas arise from convex sets and convex functions, with extensions to broader contexts.

A convex set is a subset of a real vector space such that any line segment between two points in the set lies entirely within the set. This simple property leads to important constructions such as the convex hull, the smallest convex set containing a given set, and to geometric objects like convex polytopes and balls. Convexity is preserved under intersection and under many linear operations.

A convex function is defined on a convex domain by the inequality f(tx + (1−t)y) ≤ t f(x) + (1−t) f(y) for all x, y in the domain and t in [0,1]. Equivalently, its epigraph, the set of points lying on or above its graph, is a convex set. For twice-differentiable functions, convexity is characterized by a positive semidefinite Hessian. Consequences include Jensen’s inequality and the fact that, on a convex domain, local minima are global minima.

Convex optimization studies problems of minimizing a convex objective over a convex feasible set. These problems have favorable properties, such as unique solutions under strong convexity and efficient algorithms. Variants include strong convexity, uniform convexity, and smooth convexity, which refine guarantees on existence, uniqueness, and convergence rates.

Geodesic convexity generalizes these ideas to manifolds and metric spaces by requiring that the function behaves convexly along geodesics. Convexities find applications in economics, machine learning, control, and computational geometry, where tractable optimization and stable behavior are valued.

See also: convex analysis, Jensen’s inequality, subgradient.