Packing minima of convex bodies

TL;DR

This paper introduces and studies packing minima for convex bodies, providing new volume inequalities and explicit calculations for specific convex shapes, extending previous concepts related to lattice packings.

Contribution

It generalizes the concept of packing minima to all convex bodies containing the origin and establishes new volume inequalities and explicit values for special cases.

Findings

Derived novel volume inequalities for packing minima.

Calculated packing minima for specific convex bodies.

Extended the concept to all convex bodies with the origin in their interior.

Abstract

In 2021, Henk, Schymura and Xue introduced packing minima, associated with a convex body and a lattice, as packing counterparts to the covering minima of Kannan and Lov\'asz. Motivated by conjectures on the volume inequalities for the successive minima, we generalized the definition of the packing minima to the class of all convex bodies that contain the origin in their interior. For these packing minima, we presented several novel volume inequalities and calculated the specific values of the packing minima for several special convex bodies.