epäkonveksista

The term epäkonveksista originates from the Finnish language and is the partitive form of epäkonveksinen, meaning “non‑convex”. In mathematical contexts it is used to describe objects, functions, sets or regions that do not satisfy the usual definition of convexity. A set S in a real vector space is convex if, for any two points x and y in S, the entire line segment joining x and y also lies in S. When this property fails for some pair of points, the set is called non‑convex or, in Finnish, epäkonveksinen.

Non‑convexity arises frequently in optimization, economics, and computer science. Optimization problems with non‑convex feasible sets or objective functions can have multiple local optima, making them harder to solve than convex problems, which have the desirable property that any local optimum is also a global optimum. In control theory, non‑convex reachable sets reflect the complexity of a system’s dynamics. In robotics, the configuration space of a robot arm with obstacles often produces an epäkonveksinen region that must be navigated by motion‑planning algorithms.

Geometric examples of non‑convex shapes include a crescent moon shape, a star‑shaped polygon, or any shape with a re‑entrant corner or interior indent. Algebraic examples include certain polynomial level sets or surfaces defined by equations that produce holes or isolated components. In topology, the term may also refer more generally to spaces that lack a convex structure, regardless of the metric used.

Because non‑convex spaces lack the “nice” properties of convex sets, many computational geometry algorithms are designed to either approximate a non‑convex set by a convex hull or to partition it into convex pieces. This strategy simplifies analysis and algorithmic handling, although it can incur additional approximation error.