Title: Discrete Geometry in Minkowski Spaces
Abstract:
In recent decades, many papers appeared in which typical problems of Discrete Geometry are investigated, but referring to the more general setting of finite dimensional real Banach spaces (i.e., to Minkowski Geometry). In several cases such problems are investigated in the even more general context of spaces with so-called asymmetric norms (gauges).
In many cases the extension of basic geometric notions, needed for posing these problems in non-Euclidean Banach spaces, is already interesting enough.
Examples of such notions and problems are: circumballs and -centers of convex sets (e.g., studying Chebyshev sets), corresponding inballs and -centers, packings and coverings (for instance, Lebesgue's universal covering problem), problems from Location Science (like minsum hyperplanes and minsum hyperspheres), properties of curves and surfaces
in the spirit of Discrete Differential Geometry (e.g., for polyhedral norms),
reduced and complete sets (also for polyhedral norms), applications of notions from Combinatorial Geometry (such as Helly dimension), and generalized theorems from incidence geometry (e.g., theorems of Clifford and Miquel). In this talk, an overview to several such problems and related
needed notions is given.