Almost Simplicial Polytopes: The Lower and Upper Bound Theorems

Eran Nevo, Guillermo Pineda-Villavicencio, Julien Ugon, David Yost

Research output: Contribution to journalArticlepeer-review


We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.

Original languageEnglish
Pages (from-to)537-556
Number of pages20
JournalCanadian Journal of Mathematics
Issue number2
StatePublished - 1 Apr 2020


  • Lower Bound theorem
  • Upper Bound theorem
  • almost simplicial polytope
  • f-vector
  • graph rigidity
  • h-vector
  • polytope
  • simplicial polytope

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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