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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Affine hemispheres of elliptic type

Author: B. Klartag
Original publication: Algebra i Analiz, tom 29 (2017), nomer 1.
Journal: St. Petersburg Math. J. 29 (2018), 107-138
MSC (2010): Primary 53A15, 52A20
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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Additional Information

B. Klartag
Affiliation: Department of Mathematics Weizmann Institute of Science Rehovot 7610001 Israel; School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel

Keywords: Affine sphere, cone measure, anchor, Santal\'o point, obverse
Received by editor(s): December 13, 2015
Published electronically: December 27, 2017
Additional Notes: Supported by a grant from the European Research Council
Dedicated: Dedicated to Yuri Burago at the occasion of his 80th birthday
Article copyright: © Copyright 2017 American Mathematical Society

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