Non-uniform packings

TL;DR

This paper extends the classical packing problem to include balls of varying sizes, establishing bounds on the maximum number that can fit without overlap based on average radius.

Contribution

It introduces a generalized concept of packing with non-uniform radii and derives upper bounds related to the average radius of the balls.

Findings

Derived upper bounds for non-uniform packing numbers

Connected non-uniform packing bounds to classical packing results

Provided a framework for analyzing packings with variable radii

Abstract

We generalize the classical notion of packing a set by balls with identical radii to the case where the radii may be different. The largest number of such balls that fit inside the set without overlapping is called its {\em non-uniform packing number}. We show that the non-uniform packing number can be upper-bounded in terms of the {\em average} radius of the balls, resulting in bounds of the familiar classical form.