Abstract:
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 $K$.
Keywords:affine sphere, cone measure, anchor, Santaló point, obverse.
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