Minimal pairs of convex sets which share a recession cone

TL;DR

This paper investigates minimal pairs of unbounded convex sets sharing a recession cone, providing formulas, criteria, and applications, including a generalization of a polytopal summand criterion and minimal representations of dc-functions.

Contribution

It introduces new minimal pair characterizations, formulas, and criteria for unbounded convex sets, extending existing theories and applying them to dc-functions.

Findings

Minimal pairs with translation property are reducible.

A formula for minimal pairs in 2D is provided.

The polytopal summand criterion is generalized to unbounded sets.

Abstract

Robinson introduced a quotient space of pairs of convex sets which share their recession cone. In this paper minimal pairs of unbounded convex sets, i.e. minimal representations of elements of Robinson's spaces are investigated. The fact that a minimal pair having property of translation is reduced is proved. In the case of pairs of two-dimensional sets a formula for an equivalent minimal pair is given, a criterion of minimality of a pair of sets is presented and reducibility of all minimal pairs is proved. Shephard--Weil--Schneider's criterion for polytopal summand of a compact convex set is generalized to unbounded convex sets. An application of minimal pairs of unbounded convex sets to Hartman's minimal representation of dc-functions is shown. Examples of minimal pairs of three-dimensional sets are given.