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

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 $ρ$-separable packing for given $ρ\geq 1$ if in every ball concentric to a packing element of $\mathcal P$ having radius $ρ$ (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 $ρ$-separable packing in $\mathbb{E}^d$ with $n$ being sufficiently large for given $ρ\geq 1$. 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. Böröczky Jr. [Monatsh. Math. 118 (1994), 41-54] from translative packings to $ρ$-separable translative packings for $ρ\geq 1$.

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