Packing and covering properties of sequences of convex bodies

TL;DR

This paper surveys the problem of packing and covering convex bodies within a convex set, including algorithmic online variants where bodies are placed sequentially without repositioning.

Contribution

It provides a comprehensive overview of packing and covering problems for convex bodies, focusing on both classical and online algorithmic approaches.

Findings

Analysis of conditions for packing and covering convex bodies.

Discussion of online algorithms for sequential placement.

Identification of open problems in convex body packing and covering.

Abstract

This paper suveys different variants of the following problem: Given a convex set $K$ and a sequence $\{C_i\}$ of convex bodies in $E^n$, is it possible to pack the sequence of bodies in $K$ or cover $K$ with the bodies? Algorithmic versions of these problems are on-line packing and on-line covering: The bodies of the sequence are given one at a time and the algorithm is to decide on the placement of the arriving body before the next body is revealed; once placed, the body cannot be moved.