indicatrix has the form of (1), where is even and not necessarily of positive Borel measure. The bodies bounded by such hypersurfaces are called generalized zonoids. The octahedron in the Euclidean space is not a generalized zonoid. From the above statement it follows that the flat metric of Minkowski space with the norm is not generated by the Blaschke–Busemann construction. Generalizations of Hilbert's fourth problem There was found the correspondence between the planar n-dimensional Finsler metrics and special symplectic forms on the Grassmann manifold в . There were considered periodic solutions of Hilbert's fourth problem : Let (M, g) be a compact locally Euclidean Riemannian manifold. Suppose that Finsler metric on M with the same geodesics as in the metric g is given. Then the Finsler metric is the sum of a locally Minkovski metric and a closed 1-form. Let (M, g) be a compact symmetric Riemannian space of rank greater than one. If F is a symmetric Finsler metric whose geodesics coincide with geodesics of the Riemannian metric g, then (M, g) is a symmetric Finsler space. The analogue of this theorem for rank-one symmetric spaces has not been proven yet. Another exposition of Hilbert's fourth problem can be found in work of Paiva. Unsolved problems Hilbert's fourth problem for non-symmetric Finsler metric has not yet been solved. The description of the metric on for which k-planes minimize the k''-area has not been given (Busemann). References Further reading Foundations of geometry 04