Packing convex bodies by cylinders

TL;DR

This paper develops bounds for packing convex bodies with cylinders, extending previous covering bounds and exploring multiple covering and packing scenarios, including analogs of Falconer's results.

Contribution

It introduces packing bounds for convex bodies with cylinders, complementing earlier covering bounds and extending to multiple coverings and packings.

Findings

Established packing bounds for convex bodies with cylinders.

Extended bounds to r-fold covering and packing scenarios.

Provided a packing analog of Falconer's results.

Abstract

In [BL] in relation to the unsolved Bang's plank problem (1951) we obtained a lower bound for the sum of relevant measures of cylinders covering a given d-dimensional convex body. In this paper we provide the packing counterpart of these estimates. We also extend bounds to the case of r-fold covering and packing and show a packing analog of Falconer's results ([Fa]).