Affine hemispheres of elliptic type

We find that for any $n$-dimensional, compact, convex set $K\subseteq\mathbb R^{n+1}$ there is an affinely-spherical hypersurface $M\subseteq\mathbb R^{n+1}$ with center in the relative interior of $K$ such that the disjoint union $M\cup K$ 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 Santaló point of $K$.