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Given a centrally symmetric convex body 
$B$ in 
${{\mathbb{E}}^{d}}$, we denote by 
${{\mathcal{M}}^{d}}\left( B \right)$ the Minkowski space (i.e., finite dimensional Banach space) with unit ball $B$. Let 
$K$ be an arbitrary convex body in ${{\mathcal{M}}^{d}}\left( B \right)$. The relationship between volume 
$V\left( K \right)$ and the Minkowskian thickness (= minimal width) 
${{\Delta }_{B}}\left( K \right)$ of 
$K$ can naturally be given by the sharp geometric inequality 
$V\left( K \right)\ge \alpha \left( B \right)\cdot {{\Delta }_{B}}{{\left( K \right)}^{d}}$, where 
$\alpha \left( B \right)>0$. As a simple corollary of the Rogers-Shephard inequality we obtain that 
${{\left( _{d}^{2d} \right)}^{-1}}\le \alpha \left( B \right)/V\left( B \right)\le {{2}^{-d}}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for 
$d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.
