Extremal functions on moduli spaces and applications

TL;DR

This paper investigates extremal functions related to convex bodies and their associated moduli spaces, focusing on lattice points, packing, covering, and Diophantine approximation, with specific results on Minkowski balls and domains.

Contribution

It introduces new extremal functions connecting convex geometry, lattice theory, and algebraic geometry, and determines minimal areas of inscribed and circumscribed hexagons for Minkowski bodies.

Findings

Determined minimal areas of inscribed and circumscribed hexagons for 2D Minkowski balls.

Established bounds for lattice point distributions on Minkowski curves.

Analyzed extremal functions for optimal packings and coverings.

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.