konveksse

Konveksse is a term used in mathematics to denote the property of convexity, appearing as a noun form in some languages where the standard term is convexity. In English, convexity is the ordinary term. A set S in a real or complex vector space is called konveksse if for any x and y in S and any t with 0 ≤ t ≤ 1, the point t x + (1−t) y also lies in S. Equivalently, S contains the entire line segment between any pair of its points. The concept extends to functions: a real-valued function f defined on a convex domain is convex if its epigraph is a convex set, or, equivalently, f(t x + (1−t) y) ≤ t f(x) + (1−t) f(y) for all x, y and t in [0, 1].

Key properties include: the intersection of any family of konveksse sets is konveksse; the convex hull of a set is the smallest konveksse set containing it; affine images preserve konveksse; the Minkowski sum of konveksse sets is konveksse. Examples of konveksse sets are discs, rectangles, and most polygons with straight-line boundaries; non-convex shapes include crescents, annuli, and horseshoe shapes.

Applications of konveksse arise notably in convex optimization, where minimizing a convex function over a convex feasible region guarantees global optimality under mild conditions. The concept also informs economic theory, computational geometry, and machine learning, where convexity aids in analysis and algorithm design. Related terms include convex hull, convex function, and convex set.