
Journal of Convex Analysis 26 (2019), No. 4, 12771296 Copyright Heldermann Verlag 2019 Stability Result for the Extremal Grünbaum Distance Between Convex Bodies Tomasz Kobos Faculty of Mathematics and Comp. Science, Jagiellonian University, 30348 Krakow, Poland Tomasz.Kobos@im.uj.edu.pl [Abstractpdf] In 1963 Gr\"unbaum introduced the following variation of the BanachMazur distance for arbitrary convex bodies $K, L \subset \mathbb{R}^n$: $$ d_G(K, L) = \inf \{ r \ : \ K' \subset L' \subset rK' \} $$ with the infimum taken over all nondegenerate affine images $K'$ and $L'$ of $K$ and $L$ respectively. In 2004 Gordon, Litvak, Meyer and Pajor proved that the maximal possible distance is equal to $n$, confirming the conjecture of Gr\"unbaum. In 2011 Jim\'{e}nez and Nasz\'{o}di asked if the equality $d_G(K, L)=n$ implies that $K$ or $L$ is a simplex and they proved it under the additional assumption that one of the bodies is smooth or strictly convex. The aim of the paper is to give a stability result for a smooth case of the theorem of Jim\'{e}nez and Nasz\'{o}di. We prove that for each smooth convex body $L$ there exists $\varepsilon_0(L) >0$ such that if $d_G(K, L) \geq (1\varepsilon)n$ for some $0 \leq \varepsilon \leq \varepsilon_0(L)$, then $d(K, S_n) \leq 1 + 40n^3r (\varepsilon)$, where $S_n$ is the simplex in $\mathbb{R}^n$, $r(\varepsilon)$ is a specific function of $\varepsilon$ depending on the modulus of the convexity of the polar body of $L$ and $d$ is the usual BanachMazur distance. As a consequence, we obtain that for arbitrary convex bodies $K, L \subset \mathbb{R}^n$ their BanachMazur distance is less than $n^2  2^{22}n^{7}$. Keywords: BanachMazur distance, Gr\"unbaum distance, convex body, stability, John's decomposition. MSC: 52A40, 52A20, 52A27 [ Fulltextpdf (166 KB)] for subscribers only. 