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Application of an idea of Voronoĭ to John type problems. (English) Zbl 1144.52003

By associating with each positive definite quadratic form on Euclidean space \({\mathbb E}^d\) the vector of the entries of the corresponding matrix, the set of these quadratic forms is mapped to a convex cone in \({\mathbb E}^{\frac{1}{2}d(d+1)}\). Exploiting an idea that goes back to Voronoi, the author uses this cone of positive definite quadratic forms and its closure to obtain various new John type results. These concern ellipsoids inscribed or circumscribed to convex bodies and satisfying extremum properties, and corresponding special positions of convex bodies.
For example, a John type characterization is given for minimum ellipsoidal shells of symmetric convex bodies. For typical convex bodies (in the Baire category sense), the uniqueness of minimum ellipsoidal shells is shown and the number of contact points is determined. For circumscribed ellipsoids of minimum surface area the uniqueness is shown (which is much harder than in the volume case), and the corresponding minimum positions are studied. Also studied are John type characterizations of minimum positions of a convex body with respect to (polar) moments, and a characterization of minimum \(MM^*\)-positions is obtained. Several further results are contained in this thorough investigation.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A41 Convex functions and convex programs in convex geometry
52A40 Inequalities and extremum problems involving convexity in convex geometry
46B07 Local theory of Banach spaces
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