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The number of polytopes, configurations and real matroids

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Noga Alon
Noga Alon
Affiliation:
Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel 69978.
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Abstract

We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most , as n/d → ∞. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elements of rank d. Our main tool is a theorem of Milnor and Thorn from real algebraic geometry.

MSC classification

Secondary: 52A37: Other problems of combinatorial convexity
Type
Research Article
Information
Mathematika , Volume 33 , Issue 1 , June 1986 , pp. 62 - 71
Copyright
Copyright © University College London 1986

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