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It is known that each symmetric stable distribution in $R^d$ is related to anorm on $R^d$ that makes $R^d$ embeddable in $L_p([0,1])$. In case of amultivariate Cauchy distribution the unit ball in this norm corresponds is thepolar set to a convex set in $R^d$ called a zonoid. This work interpretsgeneral stable laws using convex or star-shaped sets and exploits recentadvances in convex geometry in order to come up with new probabilistic resultsfor multivariate stable distributions. In particular, it provides expressionsfor moments of the Euclidean norm of a stable vector, mixed moments and variousintegrals of the density function. It is shown how to use geometricinequalities in order to bound important parameters of stable laws. It is shownthat each symmetric stable laws appears as the limit for the sum ofsub-Gaussian laws and an estimate for the probability distance to asub-Gaussian law is given. Operations with convex sets induce the well-knownand new operations with stable vectors. Furthermore, covariation, regressionand orthogonality concepts for stable laws acquire geometric interpretations. Asimilar collection of results is presented for one-sided stable laws.

Convex and star-shaped sets associated with stable distributions