
TL;DR
This paper establishes the existence and uniqueness of affine hemispheres of elliptic type associated with any compact convex set in R^{n+1}, linking them to the Santalo point and affine invariance.
Contribution
It introduces affine hemispheres of elliptic type, proving their existence, uniqueness, and affine invariance for any compact convex set in R^{n+1}.
Findings
Affine hemispheres are uniquely determined by the convex set.
They are centered at the Santalo point.
The construction is affine-invariant.
Abstract
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.
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