We find that for an}' n-dimensional. compact, convex set А" С R" + 1 there is an affinely-spherical hypersurface Л/ С R" ' with center in the relative interior of A' such that the disjoint union M U К is the boundary of an (n+ l)-dimensional, compact, convex sot. <...> This so-called affair hemisphere M is uniquely determined by К up to affine transformations, it is of elliptic type, is associated with A' in an affinely-invariant manner, and it is centered at the Santalo point of A"! <...>