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Register a new userOn the Facets of the Secondary Polytope
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Authors
Sven Herrmann
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The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are the simplest examples of such facets and the first that were systematically investigated. The present paper can be seen as a continuation of these studies and as a starting point of an examination of the subdivisions corresponding to the facets of the secondary polytope in general. As a special case, the notion of k-split is introduced as a possibility to classify polytopes in accordance to the complexity of the facets of their secondary polytopes. An application to matroid subdivisions of hypersimplices and tropical geometry is given.
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This paper is devoted to the study of lower and upper bounds for the number of vertices of the polytope of $ntimes ntimes n$ stochastic tensors (i.e., triply stochastic arrays of dimension $n$). By using known results on polytopes (i.e., the Upper and Lower Bound Theorems), we present some new lower and upper bounds. We show that the new upper bound is tighter than the one recently obtained by Chang, Paksoy and Zhang [Ann. Funct. Anal. 7 (2016), no.~3, 386--393] and also sharper than the one in Linial and Lurias [Discrete Comput. Geom. 51 (2014), no.~1, 161--170]. We demonstrate that the analog of the lower bound obtained in such a way, however, is no better than the existing ones.
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