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Characteristic properties of ellipsoids and convex quadrics. (English) Zbl 1433.52001
This article is a detailed and exhaustive survey of geometric characterisation of ellipsoids and convex quadrics. The type of conditions considered in the paper, according to the words of the author: “deals with a variety of intuitively clear and attractive geometric arguments, which can hardly be formalized in analytic terms”. The material is divided into several sections; in the first one, characterisation theorems of convex quadrics among convex hypersurfaces are presented. Each of the subsequent sections concerns a specific type of geometric property which may be used to characterise ellipsoids among convex bodies (compact and convex subsets of \(\mathbb{R}^n\)), or among convex solids (closed convex subsets of \(\mathbb{R}^n\)), or to characterise convex quadrics among convex hypersurfaces. More precisely, the following types of conditions are considered:
elliptic planar sections through a given point, or parallel to a given line;
symmetry of planar sections;
equivalence of parallel sections up to an homothety;
equivalence of parallel sections up to an affine transformation;
elliptic parallel and central projections;
hyperplanarity and local hyperplanarity of midsurfaces and \(\lambda\)-surfaces;
hyperplanarity of shadow boundaries (including connections with hyperplanarity of midsurfaces);
hyperplanarity of shadow boundaries by parallel or point source illumination;
hyperplanarity of intersection of the boundary with the boundary of homothetic copies;
various types of conditions, including hyperplanarity, of projective centres;
invariance under affine or projective transformations.
Each section is completed by the most recent extensions and generalisations of classical results, and related open problems. For the majority of the presented results the proof is not included, but references to the literature are detailed and complete. The list of references constitutes by itself a rich and useful tool on this interesting topic.
MSC:
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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