Selected Applications of Linear Semi-Infinite Systems Theory

In this paper we, firstly, review the main known results on systems of an arbitrary (possibly infinite) number of weak linear inequalities posed in the Euclidean space Rn (i.e., with n unknowns), and, secondly, show the potential power of this theoretical tool by developing in detail two significant applications, one to computational geometry: the Voronoi cells, and the other to mathematical analysis: approximate subdifferentials, recovering known results in both fields and proving new ones. In particular, this paper completes the existing theory of farthest Voronoi cells of infinite sets of sites by appealing to well-known results on linear semi-infinite systems. This research was partially supported by PGC2018-097960-B-C22 of the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI), and the European Regional Development Fund (ERDF); by CONICET, Argentina, Res D No 4198/17; and by Universidad Nacional de Cuyo, Secretaría de Investigación, Internacionales y Posgrado (SIIP), Res. 3922/19-R, Cod.06/D227, Argentina.