Ikkekonveks

Ikkekonveks is a term used in geometry and mathematical optimization to describe shapes, regions, or functions that are not convex. In the most common usage, a set S in a Euclidean space is convex if, for any two points x and y in S, the line segment joining x and y is entirely contained in S. If there exists at least one pair of points in S for which the segment [x,y] leaves S, then S is non-convex.

Non-convexity occurs in many familiar shapes. In two dimensions, polygons with indentations (reflex vertices) or holes, such as crescent shapes or donut-like rings, are non-convex. The annulus, the region between two concentric circles, is a classic example: two points on opposite sides may be connected by a line that passes through the empty interior. Non-convex sets can be connected or disconnected.

In higher dimensions, non-convexity generalizes similarly; a region in space may fail the segment condition or have multiple cavities and tunnels. The convex hull of a non-convex set—the smallest convex set containing it—captures the idea of “filling in” the gaps.

In optimization and machine learning, non-convex problems pose greater challenges than convex ones, because local optima do not guarantee a global optimum. This leads to a focus on global optimization methods, heuristics, or problem-specific structure to find good solutions.

Common related concepts include convexity, non-convex optimization, and quasi-convexity. In two dimensions, simple tests for non-convexity often involve examining interior angles or ensuring that the line segment between boundary points remains inside the shape.