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 $K, L \subset \mathbb{R}^n$: $d_G(K, L) = \inf \{ |r| \ : \ K' \subset L' \subset rK' \}$ with the infimum taken over all non-degenerate 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 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 $n^2 - 2^{-22}n^{-7}$.