A note on the $M^*$--limiting convolution body

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 $\ell_1^n$.