Konveksius

Konveksius is a theoretical construct in geometry and optimization that generalizes the idea of convexity. It refers to a family of sets in a real vector space that are closed under a generalized notion of convex combination, called Konveksius combinations. The Konveksius hull of a set S is defined as the smallest Konveksius set containing S, serving as a generalized analogue of the convex hull.

The concept is designed so that ordinary convexity is recovered as a special case. When the chosen rule for combining points reduces to standard weighted averages with nonnegative weights summing to one, the Konveksius hull coincides with the ordinary convex hull. By selecting different combination rules, one can capture a spectrum of generalized convexities, including variants that appear in non-convex optimization and computational geometry.

Key properties of Konveksius sets typically include stability under the Konveksius combination operation, monotonicity with respect to set inclusion, and compatibility with linear mappings. The hull operator is extensive, idempotent, and monotone, mirroring the basic axioms of hull constructions, while allowing broader closure requirements than classical convexity.

Applications of the Konveksius framework appear in educational contexts to illustrate how diverse convexity notions relate, and in theoretical studies of generalized convex optimization, approximation algorithms, and computational geometry. It also provides a unifying language for examining how relaxing or modifying convexity impacts hull constructions and associated algorithms.

See also: convex hull, generalized convexity, quasi-convexity, star-convexity, convex optimization.