Uniqueness of radial centers of parallel bodies

We show the uniqueness of the radial centers of any order $\alpha$ of a parallel body of a convex body $\mathrm{\Omega }$ in ${\mathbb{R}}^m$ at distance $\delta$ if $\delta$ is greater than the diameter of $\mathrm{\Omega }$ multiplied by a constant which depends only on the dimension $m$.