Representability of convex sets by analytical linear inequality systems

Motzkin decomposition of closed convex sets

“…(ii) If F is M-decomposable and contains an extreme point, then (11) is the minimal Motzkin representation of F , with…”

“…One says that σ is an ordinary linear system if T is finite and it is a linear semi-infinite system (an LSIS in short) otherwise. LSISs have been studied from the point of view of existence of solutions, redundancy, and the geometry of F (see, e.g., [9,11,7], and references therein).…”

Voronoi cells via linear inequality systems

“…A series of works published in the 1990s analyze other types of systems with the same purpose (see, e.g., [10] and references therein). Moreover, the theory of linear semi-in…nite systems is a well-known tool of optimization with in…nitely many linear constraints; it applies in statistics ( [2]), variational inequalities ( [18]), convex geometry ( [12]), and moment problems ( [16]). This paper presents a new application to computational geometry.…”

Selected Applications of Linear Semi-Infinite Systems Theory

“…The chapters devoted to LSIS in the monograph [46] on LSIP describe the state-of-the-art on LSIS at the end of the 20th Century. The few papers on deterministic LSIS published since 1998 deal with the representation of closed convex sets by means of particular types of LSIS ( [1], [40], [60], [37]), while the many contributions to uncertain LSIS in this period cover a variety of topics, as the continuity properties of the solution set and associated maps ( [14], [51], [42], [43]), formulas involving error bounds and di¤erent types of distances and Lipschitz-like moduli ( [13], [10], [15], [12], [11], [16], [17], [18], [61]), as well as radius of robust feasibility [38].…”

“…The recent survey [47] updates the known connections between LSIP and LSIS theories, and sketches some applications of both disciplines, in the case of LSIS to the computation of the radii of robust feasibility for uncertain linear and conic optimization problems, to polarity of closed convex sets [60] and to moment problems [66], [64], among others. Generally speaking, these applications use the LSIS machinery to prove some results in an easy way, but they do not exploit this tool systematically.…”