In 1963 Grünbaum introduced the following variation of the Banach-Mazur distance for arbitrary convex bodies
K, L \subset \mathbb{R}^n
K
,
L
⊂
R
n
:
d_G(K, L) = \inf \{ |r| : K' \subset L' \subset rK' \}
d
G
(
K
,
L
)
=
inf
{
∣
r
∣
:
K
′
⊂
L
′
⊂
r
K
′
}
with the infimum taken over all non-degenerate affine images
K'
K
′
and
L'
L
′
of
K
K
and
L
L
respectively. In 2004 Gordon, Litvak, Meyer and Pajor proved that the maximal possible distance is equal to
n
n
, confirming the conjecture of Grünbaum. In 2011 Jiménez and Naszódi asked if the equality
d_G(K, L)=n
d
G
(
K
,
L
)
=
n
implies that
K
K
or
L
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 Jiménez and Naszódi. We prove that for each smooth convex body
L
L
there exists
\varepsilon_0(L) >0
ε
0
(
L
)
>
0
such that if
d_G(K, L) \geq (1-\varepsilon)n
d
G
(
K
,
L
)
≥
(
1
−
ε
)
n
for some
0 \leq \varepsilon \leq \varepsilon_0(L)
0
≤
ε
≤
ε
0
(
L
)
, then
d(K, S_n) \leq 1 + 40n^3r (\varepsilon)
d
(
K
,
S
n
)
≤
1
+
40
n
3
r
(
ε
)
, where
S_n
S
n
is the simplex in
\mathbb{R}^n
R
n
,
r(\varepsilon)
r
(
ε
)
is a specific function of
\varepsilon
ε
depending on the modulus of the convexity of the polar body of
L
L
and
d
d
is the usual Banach-Mazur distance. As a consequence, we obtain that for arbitrary convex bodies
K, L \subset \mathbb{R}^n
K
,
L
⊂
R
n
their Banach-Mazur distance is less than
n^2 - 2^{-22}n^{-7}
n
2
−
2
−
22
n
−
7
.