konvex

Konvex is the term used in several languages to describe concepts related to convexity in mathematics. In geometry and analysis, an object is konvex if any two points within it can be joined by a line segment that lies entirely inside the object. This idea can apply to sets in a vector space as well as to functions defined on such sets.

For a subset C of a real vector space, C is konvex if for all x and y in C and all t in the closed interval [0,1], the point t x + (1−t) y also belongs to C. This means the set contains all convex combinations of its points. The smallest konvex superset of a given set is its konvex hull. Common examples of konvex sets include disks, rectangles, and polygons with straight edges; non-convex examples include crescent shapes or annuli, where a line segment between two points may leave the set.

A function f defined on a convex domain is konvex if its epigraph is a konvex set, equivalently if for all x, y in the domain and t in [0,1], f(t x + (1−t) y) ≤ t f(x) + (1−t) f(y). Strict convexity strengthens this inequality. Convexity leads to useful properties such as Jensen’s inequality and the fact that the local minimum of a convex function on a convex set is also a global minimum.

In optimization and economics, konvexity underpins convex optimization, where objective functions and feasible regions are konvex, ensuring tractable problems with well-behaved minima. Related topics include convex hulls, separation theorems, and the study of konvex bodies and polytopes.