Cells in the box and a hyperplane

  • Imre Bárány

    Renyi Institute of Mathematics, Budapest, Hungary; University College London, UK
  • Péter Frankl

    Renyi Institute of Mathematics, Budapest, Hungary
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Abstract

It is well known that a line can intersect at most cells of the chessboard. Here we consider the high-dimensional version: how many cells of the -dimensional box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact: if are convex bodies in and , then the surface area of is smaller than that of .

Cite this article

Imre Bárány, Péter Frankl, Cells in the box and a hyperplane. J. Eur. Math. Soc. 25 (2023), no. 7, pp. 2863–2877

DOI 10.4171/JEMS/1252