A note on the $M^{\ast }$--limiting convolution body

Abstract

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 $\delta$--$M^{\ast }$--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 $\delta$--$M^{\ast }$--convolution body of the $n$--dimensional cube is homothetic to the unit ball of ${\ell }_1^n$.