konvekslii

Konvekslii is a mathematical concept referring to convexity as applied to sets and functions, commonly used in geometry, analysis, and optimization. In geometry, a set S in Euclidean space is convex if for any two points x and y in S the line segment joining them lies entirely in S. Equivalently, for all x,y in S and all t in [0,1], the point tx + (1−t)y belongs to S. The convex hull of a set is the smallest convex set containing it and can be described as the set of all finite convex combinations of points from the original set.

In analysis, a function f defined on a convex domain D is convex if for all x,y in D and t in [0,1], f(tx+(1−t)y) ≤ t f(x) + (1−t) f(y). If the inequality is strict for distinct points, f is strictly convex. Convexity has useful properties: intersections of convex sets are convex, linear images of convex sets are convex, and Minkowski sums of convex sets are convex. Foundational results include separation theorems and Carathéodory’s theorem, which have wide-ranging implications in theory and computation.

Applications of konvekslii abound in optimization, economics, machine learning, and computational geometry. Convex optimization, in particular, enjoys favorable guarantees: many problems admit globally optimal solutions that can be found efficiently.

Name note: konvekslii is a transliteration used in some Slavic-language contexts for the term “convexity.” Its linguistic root is the Latin convexus.