  We introduce the mixed convolution bodies of two convex symmetric bodies. We
prove that if the boundary of a body $K$ is smooth enough then as $\delta$
tends to $1$ the $\delta$--$M^*$--convolution body of $K$ with itself tends to
a multiple of the Euclidean ball after proper normalization. On the other hand
we show that the $\delta$--$M^*$--convolution body of the $n$--dimensional cube
is homothetic to the unit ball of $\ell_1^n$.
