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- Title
Cubes and their centers.
- Authors
Thornton, R.
- 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 [6] about such sets in the plane. We find sharp estimates for the possible packing and box-counting dimensions for 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 Haussdorff dimension and settle an analogous question for the dual polytope of the cube, the orthoplex.
- Subjects
CUBES; CENTER (Mathematics); MATHEMATICAL bounds; HYPERGRAPHS; FRACTAL dimensions; SET theory
- Publication
Acta Mathematica Hungarica, 2017, Vol 152, Issue 2, p291
- ISSN
0236-5294
- Publication type
Article
- DOI
10.1007/s10474-017-0729-z