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Affine hemispheres of elliptic type. (English) Zbl 1396.53013

St. Petersbg. Math. J. 29, No. 1, 107-138 (2018) and Algebra Anal. 29, No. 1, 145-188 (2017).
For each cone in \(\mathbb R^{n+1}\), there exists a complete affine hypersphere of hyperbolic type asymptotic to it. On the other hand, complete affine hyperspheres of elliptic or parabolic types are necessarily hyperquadrics. In this paper, the author defines the concept of affine hemispheres, which are (incomplete) affine hyperspheres \(M\) of elliptic type, locally strongly convex, whose boundary is contained in a hyperplane \(L\) and encloses a convex region \(K\subset L\), called the anchor of \(M\), such that \(K\cup M\) is the boundary of the convex region \(\tilde K\subset\mathbb R^{n+1}\).
By using techniques of convex geometry, the author proves that given a compact convex set \(K\) contained in a hyperplane \(L\), there exists an affine hemisphere \(M\) with anchor \(K\), unique up to an affine transformation. Moreover, the centre of \(M\) is the Santaló point of \(K\). This result shows that, although complete affine spheres of elliptic type are rare, there are plenty of affine hemispheres of elliptic type.

MSC:

53A15 Affine differential geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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