Geodesic
Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums
Adiprasito, Karim A.1  ; Sanyal, Raman2 
1 Einstein Institute for Mathematics, Hebrew University of Jerusalem Jerusalem Israel
2 Fachbereich Mathematik und Informatik, Freie Universität Berlin Berlin Germany
Publications Mathématiques de l'IHÉS, Tome 124 (2016), pp. 99-163

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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.

DOI : 10.1007/s10240-016-0083-7
Keywords: Simplicial Complex, Local Cohomology, Simplicial Polytopes, Local Cohomology Module, Bound Theorem

Adiprasito, Karim A. 1 ; Sanyal, Raman 2

1 Einstein Institute for Mathematics, Hebrew University of Jerusalem Jerusalem Israel
2 Fachbereich Mathematik und Informatik, Freie Universität Berlin Berlin Germany 
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     journal = {Publications Math\'ematiques de l'IH\'ES},
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Adiprasito, Karim A.; Sanyal, Raman. Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums. Publications Mathématiques de l'IHÉS, Tome 124 (2016), pp. 99-163. doi: 10.1007/s10240-016-0083-7

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