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A Large Deviations Approach to the Geometry of Random Polytopes

Part of:General convexityGeometric probability and stochastic geometry

Published online by Cambridge University Press:  21 December 2009

D. Gatzouras and
A. Giannopoulos
D. Gatzouras
Affiliation:
Department of Mathematics, Secondary, Agricultural University of Athens, Iera Odos 75, 118 55 Athens, Greece. E-mail: gatzoura@aua.gr
A. Giannopoulos
Affiliation:
Department of Mathematics, University of Athens, Panepistimiopolis, 157 84, Athens, Greece. E-mail: apgiannop@math.uoa.gr
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Extract

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

MSC classification

Secondary:52A22: Random convex sets and integral geometry60D05: Geometric probability and stochastic geometry

Information

Type
Research Article
Information
Mathematika , Volume 53 , Issue 2 , December 2006, pp. 173 - 210
Copyright
Copyright © University College London 2006

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