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 are the reciprocal values of the outer (inner) radii of K [9]). The inner j-radii rj and r̄j of a body K are defined as the radii of largest j-balls contained in j-dimensional slices K ∩F of K, whereby the value of rj is obtained from maximizing and the value of r̄j is obtained from minimizing over the possible directions of F . The outer j-radii Rj and R̄j of a body K are defined as the radii of smallest j-balls containing the projection of K onto j-dimensional subspaces F , whereby the value of Rj is obtained from minimizing and the value of R̄j is obtained from maximizing over the possible directions of F . One should note that rd = r̄d is the usual inradius and Rd = R̄d is the usual outer radius. Moreover, it is well known [3] that R1 = r̄1 is the half width and r1 = R̄1 is the half diameter. In some Russian papers the inner radii rj are also called Bernstein diameters and the outer radii Rj Kolmogorov diameters (or sometimes Kolmogorov width). The inner radii rj of regular simplices were studied in [1]. Ball uses a well known result of John [10] in his proof, which also plays an important role in our computation of the outer radii Rj of regular simplices. Until recently we thought that besides the classical results of Steinhagen [15] and Jung [11] about the outer 1and the outer d-radii, respectively, the Rj’s of regular simplices were computed only in the case that j = d − 1 by Weißbach [16, 17]. The pretended open cases originally stimulated our work. However, on