Csirmaz, László (2014) *Structure of the entropy region.* In: Unemat3, July 28-31, 2014, Universidad Nacional de Colombia, Bogota, Colombia.

| PDF (Lecture I) 1111Kb | |

| PDF (Lecture II) 431Kb | |

| PDF (Lecture III) 1901Kb |

## Abstract

The entropies of all non-empty subsets of several jointly distributed random variables form the entropy vector; the entropy region is the collection of all entropy vectors. In this series of lectures we present some very recent results on the structure of the entropy region. Lecture I investigates the two and three variable case, defines the "ringing bells" distribution, defines linear and group-based distributions. Lecture II introduces polymatroids, proves that the closure of the entropy region is a closed convex full-dimensional cone where every internal point is entropic. It concludes by proving the correctness of the four known methods which can obtain new linear entropy inequalities. Lecture III gives a self-contained proof of F. Matus stating that the entropy region is not polyhedral. The lecture concludes with some ideas on how to show that the entropy region is not semi-algebraic.

Item Type: | Conference or Workshop Item (Lecture) |
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Keywords and phrases: | entropy region; polymatroid; Shannon entropy; polyhedral cone; entropy inequality; semi-algebraic set |

Subjects: | 52B11 $n$-dimensional polytopes 05B35 Matroids, geometric lattices 94A17 Measures of information, entropy |

Divisions: | Research Groups > Theoretical cryptography |

Last Modified: | 01 Sep 2014 14:40 |

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