  Let $K, D$ be $n$-dimensional convex bodes. Define the distance between $K$
and $D$ as $$ d(K,D) = \inf \{\lambda | T K \subset D+x \subset \lambda \cdot
TK \}, $$ where the infimum is taken over all $x \in R^n$ and all invertible
linear operators $T$. Assume that 0 is an interior point of $K$ and define $$
M(K) =\int_{S^{n-1}} \| \omega \|_K d \mu (\omega), $$ where $\mu$ is the
uniform measure on the sphere. Let $K^{\circ}$ be the polar body of $K$. We use
the difference body estimate to prove that $K$ can be embedded into $R^n$ so
that $$ M(K) \cdot M(K^{\circ}) \le C n^{1/3} \log^a n $$ for some absolute
constants $C$ and $a$. We apply this result to show that the distance between
two $n$-dimensional convex bodies does not exceed $n^{4/3}$ up to a logarithmic
factor.
