Approximation of convex bodies by inscribed simplices of maximum volume (Q717836)

Let \(C^n\) denote the family of all convex bodies in Euclidean \(n\)-dimensional space \(E^n\). The Banach-Mazur distance of bodies \(C_1,C_2\in C^n\) is the number \(\delta(C_1,C_2= \underset{t,h_\lambda}{}{\text{inf}} \{\lambda; t(C_2)\subset C_1\subset h_\lambda(t(C_2))\}\), where \(t\) stands for an affine transforamtion, and \(h_\lambda\) stands for any homothety with a positive ratio \(\lambda\). The Banach-Mazur distance between an arbitrary convex body and a simplex in \(E^n\) is at most \(n+2\). The author obtains this estimate as a consequence of his Theorem. For arbitrary \(C\in{\mathcal C}^h\) and for any simplex \(S\) of maximum volume contained in \(C\) we have \(C\subset (h+2)S\). For \(n\geq 2\) the ratio \((n+2)\) is best possible. It is remarked that in general, this ratio cannot be improved, as it follows from the example of any double-cone. The above theorem implies: Corollary. For an arbitrary \(C\in{\mathcal C}^2\) and a simplex \(S\), we have \(\delta(C,S)\leq n+2\).