Abstract
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.
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K. Adiprasito was supported by an EPDI/IPDE postdoctoral fellowship, a Minerva fellowship of the Max Planck Society, the DFG within the research training group “Methods for Discrete Structures” (GRK1408) and by the Romanian NASR, CNCS—UEFISCDI, project PN-II-ID-PCE-2011-3-0533.
R. Sanyal was supported by European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no 247029 and by the DFG-Collaborative Research Center, TRR 109 “Discretization in Geometry and Dynamics”.
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Adiprasito, K.A., Sanyal, R. Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums. Publ.math.IHES 124, 99–163 (2016). https://doi.org/10.1007/s10240-016-0083-7
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DOI: https://doi.org/10.1007/s10240-016-0083-7