On the Perelman-Pukhov quotient of successive radii: better and asymptotically optimal bounds

Abstract

Perel'man in 1987 and independently Pukhov in 1979 proved that the quotient between the $(n-i+1)$-th successive outer radius and the $i$-th successive inner radius of a convex body in $n$-dimensions is not larger than $i+1$. Apart from the solved cases by Jung 1901 $(i=1)$ and Steinhagen 1921 $(i=n)$, only Perel'man (1987, $n=3$, $i=2$) and González Merino (2017, $n\geq 4$, $i=2$ and $i=n-1$) provided small improvements that beat this bound. In this paper, we obtain sharper inequalities using relations between these inner and outer measures with the diameter and minimal width. We improve the current bounds in the following cases: $i=3$ when $4\leq n \leq 8$, $i=4$ when $n=5$, $6$, $i=5$ when $n=6$, $i=6$ when $n=7$, and for every $i\geq n-Θ(\log n)$. Notably, our bounds provide the right order in $n$ when $i=n-m$, with $m$ constant and $n$ arbitrarily large. Additionally, we improve the case $n=5$, $i=3$ even further by refining an idea of Perel'man and using the optimal lower bound of the inradius in terms of the circumradius and the diameter in 3-space (see [7]).