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

Abstract

A packing of translates of a convex body in the d-dimensional Euclidean space \({{\mathrm{\mathbb {E}}}}^d\) 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 \({{\mathrm{\mathbb {E}}}}^d\) a \(\rho \)-separable packing for given \(\rho \ge 1\) 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 \({{\mathrm{\mathbf {Q}}}}\) of n non-overlapping translates of an arbitrary centrally symmetric convex body \(\mathbf {C}\) forming a \(\rho \)-separable packing in \({{\mathrm{\mathbb {E}}}}^d\) with n being sufficiently large for given \(\rho \ge 1\). If \({{\mathrm{\mathbf {Q}}}}\) has minimal mean i-dimensional projection for given i with \(1\le i<d\), then \({{\mathrm{\mathbf {Q}}}}\) is approximately a d-dimensional ball. This extends a theorem of Boroczky (Monatsh Math 118:41–54, 1994) from translative packings to \(\rho \)-separable translative packings for \(\rho \ge 1\).