Zonoid theory and Hilbert's fourth problem

A finite vector sum of line segments is termed a zonotope. A zonoid is a Blaschke-Hausdorff limit of zonotopes. A projective metric d on a convex subset of projective space is shown to be of negative type if and only if the spheres in any tangent space are polar duals of zonoids. It follows that metric arclength can be represented by a Crofton formula with respect to a positive measure on the hyperplanes if and only if d is of negative type. These ideas allow a nice characterization of this cone of metrics.