  In 1963 Gr\"unbaum introduced a 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 $d_G(K, S_n) \leq 1 +
40n^3r(\varepsilon)$, where $S_n$ is the simplex in $\mathbb{R}^n$ and
$r(\varepsilon)$ is a specific function of $\varepsilon$ depending on the
modulus of the convexity of the polar body of $L$. As a consequence we obtain
that for arbitrary convex bodies $K, L \subset \mathbb{R}^n$ their Banach-Mazur
distance is less than $n^2 - 2^{-22}n^{-7}$.
