Empty simplices of polytopes and graded betti numbers

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11 Scopus citations


The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope and applying a result of Migliore and the author. This approach allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.

Original languageEnglish
Pages (from-to)389-410
Number of pages22
JournalDiscrete and Computational Geometry
Issue number1-3
StatePublished - Mar 2008

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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