Affine hemispheres of elliptic type

Klartag, B.

doi:10.1090/spmj/1484

Affine hemispheres of elliptic type
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by B. Klartag

St. Petersburg Math. J. 29 (2018), 107-138

DOI: https://doi.org/10.1090/spmj/1484

Published electronically: December 27, 2017

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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 $K$.

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