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Characterization of different classes of convex bodies via orthogonality. (English) Zbl 1246.52001
The authors introduce the notion of affine orthogonality with respect to a convex body. In contrast to the concept of normal to a convex body, which is due to H. G. Eggleston [Isr. J. Math. 3, 163–172 (1965; Zbl 0166.17901)], one does not need a metric for this new definition. Using affine orthogonality, the authors characterize convex bodies of constant width in strictly convex and smooth normed planes (Theorem 4.6) and other special classes of convex bodies.
MSC:
52A10 Convex sets in \(2\) dimensions (including convex curves)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
46B20 Geometry and structure of normed linear spaces
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Full Text: Euclid