A weighted Minkowski theorem for pseudo-cones
Rolf Schneider
Abstract
A nonempty closed convex set in R n , not containing the origin, is called a pseudo-cone if with every x it also contains λx for λ ≥ 1 . We consider pseudo-cones with a given recession cone C , called C -pseudo-cones. The family of C -pseudo-cones can, with reasonable justification, be considered as a counterpart to the family of convex bodies containing the origin in the interior. For a C -pseudo-cone one can naturally define a surface area measure and a covolume. Since they are in general infinite, we introduce a weighting, leading to modified versions of surface area and covolume. These are finite and still homogeneous, though of other degrees. Our main result is a Minkowski type existence theorem for C -pseudo-cones with given weighted surface area measure.