Minimizing the mean projections of finite $\rho$-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 $\rho$-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 $\mathbf{Q}$ 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 $\rho \ge 1$. If $\mathbf{Q}$ has minimal mean $i$-dimensional projection for given $i$ with $1\le i<d$, 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 $\rho \ge 1$.