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Relative Stanley-Reisner theory and Uppe...
Adiprasito, Karim...

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Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums by Adiprasito, Karim ( Author )
Lemonjar Open Access
Australian National University
04-08-2023
In this paper we settle 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 faces of Minkowski sums. This has a wide range of applications and generalizes the classical the Upper Bound Theorems of McMullen and Stanley. Our main tool is relative Stanley--Reisner theory, a powerful generalization of the algebraic theory of simplicial complexes inaugurated by Hochster, Reisner, and Stanley. A key feature of our theory is the ability to accomodate topological as well as combinatorial restrictions. We illustrate this by providing several simplicial isoperimetric and reverse isoperimetric inequalities.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1405.7368
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