Stability result for the extremal Gr\"unbaum distance between convex bodies

In 1963 Gr"unbaum introduced the following variation of the Banach-Mazur distance for arbitrary convex bodies $K,L\subset {\mathbb{R}}^n$: $d_G(K,L)=\inf \{\mathrm{\mid }r\mathrm{\mid }:K^{\mathrm{\prime }}\subset L^{\mathrm{\prime }}\subset r{K}^{\mathrm{\prime }}\}$ with the infimum taken over all non-degenerate affine images $K^{\mathrm{\prime }}$ and $L^{\mathrm{\prime }}$ 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 ${\epsilon }_0(L)>0$ such that if $d_G(K,L)\ge (1-\epsilon )n$ for some $0\le \epsilon \le {\epsilon }_0(L)$, then $d(K,S_n)\le 1+40n^{3}r(\epsilon )$, where $S_n$ is the simplex in ${\mathbb{R}}^n$, $r(\epsilon )$ is a specific function of $\epsilon$ depending on the modulus of the convexity of the polar body of $L$ and $d$ is the usual Banach-Mazur distance. As a consequence, we obtain that for arbitrary convex bodies $K,L\subset {\mathbb{R}}^n$ their Banach-Mazur distance is less than $n^2-2^{-22}{n}^{-7}$.