We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most (n/d)d2n(1+0(1))as n/d→ä. 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 Thom from real algebraic geometry.
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