Radon transformations and zonoids (Q1358469)

The function \(H\) on \(\mathbb{R}^n\) is the support function of a zonoid if it is of the form \[ H(x)= \int_{S^{n-1}} \bigl| \langle x,u \rangle\bigr | d\mu(u) \] with a finite positive measure \(\mu\) on the unit sphere \(S^{n-1}\). The following theorem is proved. Let \(H\) be a subharmonic function on \(\mathbb{R}^n\) which is positively homogeneous of degree one (in particular, \(H\) can be a support function of a convex body). Then \(H\) coincides, up to a linear function, with the support function of some zonoid if and only if \[ \int_{\mathbb{R}^n} H(x)\Delta \varphi (x)dx\geq 0 \] for any \(C^\infty\) function \(\varphi\) on \(\mathbb{R}^n\) with nonnegative Radon transform.