Lectures on 0/1-polytopes

Abstract

These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an \emph{introduction} and \emph{invitation}. Rather than heading for an extensive survey on 0/1-polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1-polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e.g. using {\tt polymake}). However, any intuition that is derived from the analysis of examples in ``low dimensions'' will miss the true complexity of 0/1-polytopes. Thus, in the following we will study several aspects of the complexity of higher-dimensional 0/1-polytopes: the doubly-exponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to verify some of them on explicit examples, which are accessible as a {\tt polymake} database.

45 pages, many figures; to appear in "Polytopes - Combinatorics and Computation" (G. Kalai and G.M. Ziegler, eds.), DMV Seminars Series, Birkh"auser Basel

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