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

Karim A. Adiprasito*, Raman Sanyal

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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.

Original languageAmerican English
Pages (from-to)99-163
Number of pages65
JournalPublications Mathematiques de l'Institut des Hautes Etudes Scientifiques
Issue number1
StatePublished - 1 Nov 2016

Bibliographical note

Publisher Copyright:
© 2016, IHES and Springer-Verlag Berlin Heidelberg.


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