  We find that for any n-dimensional, compact, convex subset K of R^{n+1} there
is an affinely-spherical hypersurface M in R^{n+1} with center at the relative
interior of K, such that the disjoint union of M and K is the boundary of an
(n+1)-dimensional, compact, convex set. This so-called affine hemisphere M is
uniquely determined by K up to affine transformations, it is of elliptic type,
is associated with K in an affinely-invariant manner, and it is centered at the
Santalo point of K.
