Asymptotics of convex sets in Euclidean and hyperbolic spaces

• Autores: Igor Rivin
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 220, Nº 4, 2009, págs. 1297-1315
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.11.014
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• Resumen

  • We study convex sets C of finite (but non-zero) volume in and . We show that the intersection C8 of any such set with the ideal boundary of has Minkowski (and thus Hausdorff) dimension of at most (n-1)/2, and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C8 is a smooth submanifold of . In the hyperbolic case, we show that for any k(n-1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body in , and give asymptotic estimates as 1kn.