Publications mathématiques de l'IHÉS

, Volume 124, Issue 1, pp 99–163 | Cite as

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

  • Karim A. Adiprasito
  • Raman Sanyal


In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley.

Our main observation is that within (relative) Stanley–Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.


Simplicial Complex Local Cohomology Simplicial Polytopes Local Cohomology Module Bound Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© IHES and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Einstein Institute for MathematicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

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