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On some special surfaces connected with convex surfaces of the Lobachevskij space. (Russian, English) Zbl 1021.53009

Sib. Mat. Zh. 43, No. 5, 1020-1025 (2002); translation in Sib. Math. J. 43, No. 5, 822-825 (2002).
For a closed convex hypersurface \(F\) in the Lobachevskij \(n\)-space \((n\geq 2)\) \(E_K\) of curvature \(K<0\), the author introduces the following five parameters: the radius \(\Lambda(F)\) of the inscribed sphere; the radius \(M(F)\) of the circumscribed sphere; the maximal number \(\lambda(F)\) such that a sphere of radius \(\lambda(F)\) can be rolled freely over the inner side of \(F\); the minimal number \(M(F)\) such that \(F\) can be rolled freely over the inner side of a sphere of radius \(M(F)\); and the maximal number \(\nu(F)\) such that \(F\) can be rolled freely over the inner side of an equidistant surface of radius \(\nu(F)\). The author finds exact relations between these parameters for some classes of convex hypersurfaces in Lobachevskij spaces.

MSC:

53A35 Non-Euclidean differential geometry
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
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