Radii of Regular Polytopes

Radii of Regular Polytopes

There are three types of regular polytopes which exist in every dimension d: regular simplices, (hyper-) cubes, and regular cross-polytopes. In this paper we investigate two pairs of inner and outer j-radii (rj, Rj) and (r̄j, R̄j) of these polytopes (inner and outer radii classes are almost always considered in pairs, such that for a 0-symmetric body K and its dual K the inner (outer) radii of K ...

Regular polytopes

On Mean Outer Radii of Random Polytopes

In this paper we introduce a new sequence of quantities for random polytopes. Let KN = conv{X1, . . . ,XN} be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body K of R. We prove that the so-called k-th mean outer radius R̃k(KN ) has order max{ √ k, √ logN}LK with high probability if n ≤ N ≤ e √ . We also show that this is also the right or...

Spectra of Regular Polytopes

Introduction As is well known, the combinatorial problem of counting paths of length n between two xed vertices in a graph reduces to raising the adjacency matrix A of the graph to the n-th power ((B], p. 11). For an undirected graph, A is symmetric and the problem above simpliies considerably if its spectrum (A) is known and contains few distinct elements. Spectra of graphs, meaning spectra of...

Regular Polytopes in Z

In [3], all embeddings of regular polyhedra in the three dimensional integer lattice were characterized. Here, we prove some results toward solving this problem for all higher dimensions. Similarly to [3], we consider a few special polytopes in dimension 4 that do not have analogues in higher dimensions. We then begin a classification of hypercubes, and consequently regular cross polytopes in t...