Minimizing the mean projections of finite $\rho$-separable packings

TL;DR

This paper investigates the geometric properties of $ ho$-separable packings of convex bodies in Euclidean space, showing that minimal mean projections lead to shapes close to a sphere, extending previous results to a broader class of packings.

Contribution

It extends B"or"oczky's theorem from translative packings to $ ho$-separable packings, characterizing the shape of convex hulls with minimal mean projections.

Findings

Convex hulls of large $ ho$-separable packings are approximately spherical.

Minimal mean projections are achieved by shapes close to a Euclidean ball.

The result generalizes previous theorems to a wider class of packings.

Abstract

A packing of translates of a convex body in the $d$-dimensional Euclidean space $\mathbb{E}^d$ is said to be totally separable if any two packing elements can be separated by a hyperplane of $\mathbb{E}^{d}$ disjoint from the interior of every packing element. We call the packing $\mathcal P$ of translates of a centrally symmetric convex body $\mathbf{C}$ in $\mathbb{E}^d$ a $\rho$-separable packing for given $\rho\geq 1$ if in every ball concentric to a packing element of $\mathcal P$ having radius $\rho$ (measured in the norm generated by $\mathbf{C}$) the corresponding sub-packing of $\mathcal P$ is totally separable. The main result of this paper is the following theorem. Consider the convex hull $\mathbf{Q}$ of $n$ non-overlapping translates of an arbitrary centrally symmetric convex body $\mathbf{C}$ forming a $\rho$-separable packing in $\mathbb{E}^d$ with $n$ being sufficiently large for given $\rho\geq 1$. If $\mathbf{Q}$ has minimal mean $i$-dimensional projection for given $i$ with $1\leq i<d$, then $\mathbf{Q}$ is approximately a $d$-dimensional ball. This extends a theorem of K. B\"or\"oczky Jr. [Monatsh. Math. 118 (1994), 41-54] from translative packings to $\rho$-separable translative packings for $\rho\geq 1$.