Relationships between widths of a convex body and of an inscribed parallelotope (Q2716969)

Let \(C\subset {\mathbb R}^n\) be a convex body, let \(P\subset C\) be a parallelotope, and let \(w_i\) denote the ratio of the width of \(C\) and the width of \(P\) for the direction perpendicular to the \(i\)th pair of parallel facets of \(P\) (\(i=1,\ldots,n\)). NEWLINENEWLINENEWLINEFor \(n=2\) the author previously showed that \(1/w_1+1/w_2\geq 1\), for \(n\geq 3\) the corresponding inequality \(\sum_{i=1}^n 1/w_i\geq 1\) is an unresolved conjecture. In the present paper, the author establishes some weaker results, mainly for \(n=3\); for instance, it is proved that \(1/w_1+1/w_2+1/a_3\geq 1\), where \(a_3\leq w_3\) is a suitably defined relative axial diameter of \(C\subset {\mathbb R}^3\).