Konveks

Konveks is a term used in Danish and Norwegian to denote the mathematical concept of convexity. In geometry and analysis, a set is described as convex if, for any two points within the set, the line segment joining them lies entirely inside the set.

Formally, a subset S of a vector space is convex when for all x and y in S and all t in the interval [0, 1], the point t x + (1 − t) y also belongs to S. This simple condition implies several important properties: convex shapes have no indentations, and the straight line between any two points stays inside. A circle, a disk, and a convex polygon are examples of convex sets, while a shape with a notch or a crescent is typically non-convex.

A real-valued function f defined on a convex domain is convex if, for all x and y in its domain and all t in [0, 1], f(t x + (1 − t) y) ≤ t f(x) + (1 − t) f(y). The graph-based notion of convexity is equivalent to the convexity of the epigraph of f, the set of points lying on or above the graph. Strict convexity, uniform convexity, and other refinements describe stronger curvature properties.

The convex hull of a set is the smallest convex set containing it. In computational geometry, algorithms such as Graham scan and QuickHull compute convex hulls efficiently. Convexity underpins many fields: convex optimization studies problems where the objective and feasible region are convex; in economics, convex preferences model diminishing marginal rates; in various sciences, convex analysis provides fundamental tools.

Across languages, konveks relates to the broader, more widely used term convex, with minor spelling variations such as konvex in some contexts.