Konveksian

Konveksian, or convexity, is a fundamental concept in geometry and mathematical analysis describing a property of sets and functions that behave well under linear interpolation. The term is used across disciplines such as geometry, optimization, economics, and machine learning to denote this stable, “no-holes” structure. In many languages, konveksian is the standard term for convexity.

For sets, a subset S of a real vector space is convex if, for any two points x and y in S and any t between 0 and 1, the point tx + (1 − t)y also lies in S. Intuitively, the line segment connecting any pair of points in S stays inside S. Consequences include that the convex hull of a set—the smallest convex set containing it—is a key construction. Examples of convex sets include disks, polygons with all interior angles not exceeding 180 degrees, and more generally any polytope or half-space. Non-convex sets, such as annuli or crescent shapes, do not satisfy the condition.

For functions, a function f defined on a convex domain is convex if its graph lies below the straight line segment joining any two points on the graph, formally f(tx + (1 − t)y) ≤ t f(x) + (1 − t) f(y) for all x, y in the domain and 0 ≤ t ≤ 1. The epigraph of a convex function—the set of points above its graph—is also convex. Convexity implies that local minima are global minima and underpins powerful tools such as Jensen’s inequality and subgradient methods. Examples include f(x) = x^2 and any affine function.

Konveksian is central to convex optimization, where many problems are efficiently solvable due to these structural properties.