Volumes of Convex Lattice Polytopes and A Question of V. I. Arnold

We show by a direct construction that there are at least exp { c V ( d - 1 ) / ( d + 1 ) } convex lattice polytopes in R d of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the f...

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Bibliographic Details
Published in:Acta mathematica Hungarica 2014-10, Vol.144 (1), p.119-131
Main Authors: Bárány, I., Yuan, L.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:We show by a direct construction that there are at least exp { c V ( d - 1 ) / ( d + 1 ) } convex lattice polytopes in R d of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. This is achieved by considering the family P d ( r ) (to be defined in the text) of convex lattice polytopes whose volumes are between 0 and r d / d !. Namely we prove that for P ∈ P d ( r ) , d ! vol P takes all possible integer values between cr d–1 and r d where c > 0 is a constant depending only on d .
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-014-0418-0