TY - JOUR
T1 - Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums
AU - Adiprasito, Karim A.
AU - Sanyal, Raman
N1 - Publisher Copyright:
© 2016, IHES and Springer-Verlag Berlin Heidelberg.
PY - 2016/11/1
Y1 - 2016/11/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84961813341&partnerID=8YFLogxK
U2 - 10.1007/s10240-016-0083-7
DO - 10.1007/s10240-016-0083-7
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AN - SCOPUS:84961813341
SN - 0073-8301
VL - 124
SP - 99
EP - 163
JO - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques
JF - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques
IS - 1
ER -