Polytopes of minimal volume with respect to a shell -- another characterization of the octahedron and the icosahedron (Q2464358)

For a given real number \(r>1\), the authors consider the family \({\mathcal F}_r\) of convex bodies in the \(3\)-dimensional Euclidean space which contain the unit ball \(B^3\) and whose extreme points have distance at least \(r\) from the origin. The main result in this paper states that if \(r=\sqrt{3}\) or \(r=\sqrt{15-6\sqrt{5}}\) then the regular octahedron or icosahedron, respectively, circumscribed around \(B^3\), are the sets in \({\mathcal F}_r\) with both minimal volume and surface area. It answers a question by \textit{J. Molnár} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 30, 700--705 (1961; Zbl 0107.39901)]. Moreover they conjecture that in the case \(r=3\) the optimal figure should be a regular tetrahedron and point out that no regular polytope is extremal in its class if \(n\geq 8\). The proof of this theorem is strongly based in the fact that the optimal faces are regular triangles, which is also shown in the paper.