On weighted parallel volumes and spaces of continuous functionals of convex bodies

The functionals, that map a convex body $K$ to $\int_{[0,\infty)} < br > V_d(K+\lambda B)\, d\rho(\lambda)$, where $\rho$ is a signed measure on $[0,\infty)$ and $B$ is a fixed convex body, $V_d$ denotes the Lebesgue measure and $\mathcal{K}$ denotes the set of convex bodies, are called weighted parallel volumes. We will show that the measure $\rho$ is determined uniquely by the weighted parallel volume induced by it. The Riesz representation theorem applied to a certain generalisation of this result yields the fact that the functionals that map a (convex) body $K$ to the real number $V_d(K+A)$, where $A$ is a body, spann a dense subspace of the space of all continuous, translation-invariant functionals from the set of all (convex) bodies to the real line.