Extremal functions on moduli spaces and applications

Abstract

Our object of study is extremal functions which are defined by distance functions of convex bodies. These functions take values in the moduli spaces of algebraic and geometric objects associated with these ${\mathbb Z}$-modules (geometric lattices) and with convex bodies. In most cases, convex bodies are $2$-dimensional Minkowski balls whose boundaries are Minkowski curves and we study lattice points on these curves. We define and investigate extremal functions that yield the homogeneous arithmetic minimum of a function in a lattice, the Hermite constant, the critical determinant of a body, optimal packings of bodies, best values of covering constants, and optimal solutions of Diophantine approximation problems. Moreover, for two-dimensional unit Minkowski balls and Minkowski domains we determine the minimal areas of inscribed and circumscribed hexagons.