Cubes and their centers

Abstract

We study the relationship between the sizes of sets $B,S$ in $\mathbb{R}^n$ where $B$ contains the $k$-skeleton of an axes-parallel cube around each point in $S$, generalizing the results of Keleti, Nagy, and Shmerkin about such sets in the plane. We find sharp estimates for the possible packing and box-counting dimensions of $B$ and $S$. These estimates follow from related cardinality bounds for sets containing the discrete skeleta of cubes around a finite set of a given size. The Katona-Kruskal theorem from hypergraph theory plays an important role. We also find partial results for the Hausdorff dimension and settle an analogous question for the dual polytope of the cube, the orthoplex.