Lectures on Convex Geometry by Daniel Hug & Wolfgang Weil

Author:Daniel Hug & Wolfgang Weil
Language: eng
Format: epub, pdf
ISBN: 9783030501808
Publisher: Springer International Publishing

The final part of the proof of the Alexandrov–Fenchel inequality requires further preparations.

Recall that we denote by the set of polytopes in . We write for the subset of n-dimensional polytopes of . For vectors and , we consider polyhedral sets of the form

Clearly, if , then 0 ∈ P [h] and P [h] is a polytope if and only if the vectors are not contained in any hemisphere. Further, if and only if h 1, …, h N > 0. The vector h is called the vector of support numbers of P [h] if u 1, …, u N are the exterior unit facet normals of P [h], that is, if for i = 1, …, N. In this case, the support numbers are uniquely determined by P [h], since h(P [h], u i) = h i.

Strongly Isomorphic Polytopes

Definition 3.7

(a)A polytope is simple if each vertex of P is contained in precisely n facets of P.

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