Nemkonvex

Nemkonvex is a term used in mathematical contexts, primarily in Hungarian-language literature, to describe objects or sets that are not convex. In English, the standard term is non-convex. The adjective applies to subsets of a vector space or to certain geometric objects, indicating that the line segment between some pair of points in the set is not entirely contained within the set.

Formally, a subset S of a vector space is convex if for any x and y in S and any t in the closed interval [0,1], the point tx + (1−t)y also lies in S. If there exist x, y in S and a t in (0,1) such that tx + (1−t)y is not in S, then S is nemkonvex. This property is central to many areas of geometry and analysis, and its violation is what characterizes non-convex shapes.

Typical examples of nemkonvex shapes include crescents, annuli or donuts, L-shaped polygons, and configurations consisting of two disjoint disks or other separated components. Such shapes often have indentation or holes that prevent the containment of all line segments between their points. Non-convexity can also arise in higher dimensions and in sets defined by nonlinear constraints.

In optimization, non-convexity has important implications. Non-convex problems can have multiple local minima and maxima, making global optimization more challenging. As a result, practitioners frequently use convex relaxation, heuristics, or global-search techniques to obtain approximate solutions or bounds.

See also convexity, non-convex optimization, geometry, and related concepts in mathematical analysis.