The Minkowski problem for polytopes |
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Authors: | Daniel A. Klain |
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Affiliation: | Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, MA 01854, USA |
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Abstract: | The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski's inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context. In this article we adapt the classical argument to prove both the existence theorem of Minkowski and his mixed volume inequality simultaneously, thereby providing a new proof of Minkowski's inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry. |
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Keywords: | Polytope Isoperimetric Geometric inequality Minkowski problem Mixed volume Brunn-Minkowski Convex |
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