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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 [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.

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Correspondence to R. Thornton.

Additional information

This research was conducted as part of the Budapest Semesters in Mathematics Undergrauate Research Experience Program and was advised by Tamás Keleti.

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Thornton, R. Cubes and their centers. Acta Math. Hungar. 152, 291–313 (2017). https://doi.org/10.1007/s10474-017-0729-z

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  • DOI: https://doi.org/10.1007/s10474-017-0729-z

Key words and phrases

  • cube
  • cube skeleton
  • orthoplex
  • Hausdorff dimension
  • box-counting dimension
  • packing dimension

Mathematics Subject Classification

  • primary 05B30
  • 28A78
  • secondary 05D99
  • 52C35