convexitet

Convexitet is a fundamental concept in geometry and analysis describing how a set or a function behaves under mixing of points. In English it is convexity, and in several languages the term appears as convexitat or convexitate. The essential idea is stability under convex combinations.

Convex sets: A subset S of a vector space is convex if for any a, b in S and any t in [0,1], the point ta + (1−t)b also lies in S. Geometrically, the line segment between any two points of S stays inside S. Examples include disks, balls, and polygons without indentations; non-examples include annuli and crescent shapes.

Convex functions: A real-valued function f defined on a convex domain is convex if for all x, y in its domain and t in [0,1], f(tx + (1−t)y) ≤ t f(x) + (1−t) f(y). Equivalently, the epigraph of f, the set of points (x, r) with r ≥ f(x), is convex.

Preservation of convexity: the sum of convex functions is convex; nonnegative linear combinations preserve convexity; the pointwise maximum of a family of convex functions is convex. Composition with affine maps preserves convexity.

Convex hulls and separation: the convex hull of a set is the smallest convex set containing it. Many results in optimization rely on separating a point from a convex set with a supporting hyperplane.

Second derivative and curvature: in one variable, a twice-differentiable function is convex on an interval if f''(x) ≥ 0. In multiple variables, the Hessian must be positive semidefinite.

Applications and related notions: convexity underpins many optimization problems, economics, and machine learning, where local minima are global. Related concepts include concavity and strict or strong convexity, which strengthen the basic inequality.