## Abstract

Consider the projections of a finite set (Formula presented.) onto the coordinate hyperplanes; how small can the sum of the sizes of these projections be, given the size of A? In a different form, this problem has been studied earlier in the context of edge-isoperimetric inequalities on graphs, and it can be derived from the known results that there is a linear order on the set of n-tuples with non-negative integer coordinates, such that the sum in question is minimised for the initial segments with respect to this order. We present a new, self-contained and constructive proof, enabling us to obtain a stability result and establish algebraic properties of the smallest possible projection sum. We also solve the problem of minimising the sum of the sizes of the one-dimensional projections.

Original language | English |
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Pages (from-to) | 493-511 |

Number of pages | 19 |

Journal | Discrete and Computational Geometry |

Volume | 60 |

Issue number | 2 |

Early online date | 8 Mar 2018 |

DOIs | |

Publication status | Published - Sep 2018 |

## Keywords

- Isoperimetric problem
- Loomis–Whitney inequality
- Projections