On extensions of Minkowski's theorem on successive minima

TL;DR

This paper explores extensions of Minkowski's 2nd theorem by relaxing symmetry assumptions and replacing volume with surface area, providing new bounds in the geometry of numbers.

Contribution

It introduces novel bounds for convex bodies without symmetry and with surface area instead of volume, expanding Minkowski's theorem applicability.

Findings

Extended bounds for convex bodies with centroid at the origin

Derived surface area inequalities analogous to volume bounds

Generalized Minkowski's theorem beyond symmetric convex sets

Abstract

Minkowski's 2nd theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of a $o$-symmetric convex body in terms of its successive minima. In this paper we study extensions of this theorem from two different points of view: either relaxing the symmetry condition, assuming that the centroid of the set lies at the origin, or replacing the volume functional by the surface area.