On the convex hull of symmetric stable processes

Let $\alpha \in (1,2]$ and $X$ be an $\mathbb{R}^d$-valued symmetric $\alpha$-stable Lévy process starting at 0. We consider the closure $S_t$ of the path described by $X$ on the interval $[0,t]$ and its convex hull $Z_t$. The first result of this paper provides a formula for certain mean mixed volumes of $Z_t$ and in particular for the expected first intrinsic volume of $Z_t$. The second result deals with the asymptotics of the expected volume of the stable sausage $Z_t+B$ (where $B$ is an arbitrary convex body with interior points) as $t\to 0$. For this we assume that $X$ has independent components.