## Abstract

The colorful simplicial depth of a collection of d + 1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. [7]. Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton [6] in the theory of normal surfaces.

Original language | English |
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Article number | 184 |

Pages (from-to) | 1894-1919 |

Number of pages | 26 |

Journal | International Mathematics Research Notices |

Volume | 2019 |

Issue number | 6 |

DOIs | |

State | Published - 22 Mar 2019 |

### Bibliographical note

Publisher Copyright:© The Author(s) 2017.