As many antipodes as vertices on convex polyhedra (Q2891067)

The paper under review deals with antipodal points of convex polyhedra. By definition, given a convex polyhedron \(P\), a point \(p\in P\) is an antipode of a point \(o\in P\), if \(p\) is fastest away from \(o\) with respect to the intrinsic distance of \(P\). It is known that if \(P\) has \(n\) vertices, then any point of \(P\) has at most \(n\) different antipodes, see [\textit{J. Rouyer}, Adv. Geom. 10, No. 3, 403--417 (2010; Zbl 1217.52002)]. It turns out that this estimate is optimal.NEWLINENEWLINENamely, the authors construct a family of convex polyhedra with \(n\) vertices, \(F_n\), such that for any \(F_n\) there exists a point \(o\in F_n\) with exactly \(n\) antipodes, and all antipodes of \(o\) but one are vertices of \(F_n\). The construction is purely geometric and based on the Alexandrov gluing theorem, see [\textit{A. D. Alexandrov}, Convex polyhedra. Berlin: Springer (2005; Zbl 1067.52011)]. Moreover, in the case of \(n=4\) it is shown that \(F_n\) may be deformed so that at least two antipodes of \(o\) are non-vertex points. Finally the limit case when \(n\) tends to infinity is considered and the following statement is proved.NEWLINENEWLINETheorem. Let \(H\) be an open half-circle. Let \(K\) be a compact subset of \(H\) and \(x\) a point which does not belong to \(K\). Then there exists a convex surface \(S\) and a point \(o\in S\) such that the set of antipodes of \(o\) in \(S\), endowed with the intrinsic distance of \(S\), is locally isometric to \(K\cup\{ x\}\) endowed with the Euclidean distance of \(\mathbb{R}^2\).NEWLINENEWLINEAs corollary, for any \(\epsilon>0\) there exist a convex surface \(S\) and a point \(o\in S\) such that the length (\(1\)-dimensional Hausdorff measure) of the set of antipodes of \(o\), divided by the distance between \(o\) and one of its antipodes, is greater than \(\pi-\epsilon\). Moreover, any \(d\in [0,1]\) can be realized as the Hausdorff dimension of the set of antipodes of some point of some convex surface, cf. [\textit{T. Zamfirescu}, Math. Z. 226, No. 4, 623--630 (1997; Zbl 0896.52005)].