Approximation of general smooth convex bodies
Let \(K\) be a convex body. In the theory of polytopal approximation, one considers the polytope \(P_n\) with \(n\) vertices, or the polytope \(P_{(n)}\) with \(n\) facets closest to \(K\), and the main task is to determine the asymptotic behaviour of the distance of \(P_n\) (or \(P_{(n)})\) and \(K\). The main goal of this paper is to verify the conjecture of \textit{P. M. Gruber} [Suppl. Rend. Circ. Mat. Palermo (2) 50, 189-216 (1997; Zbl 0896.52014)], namely, to remove the positivity condition on the curvature. The two results proved in the paper are: Theorem A. The asymptotic formulae (1) to (9) (stated in the paper) hold for any convex body \(K\) with \(C^2\) boundary even if the Gauss curvature is allowed to be zero. Theorem B. If \(\partial K\) is \(C^2\) then the projections onto \(\partial K\) of the vertices of \(P_n\) (the facets of \(P_{(n)})\) are uniformly distributed with respect to the density function \(P(x)\) which appears in the asymptotic formulae (1) to (9).
scientific article; zbMATH DE number 1507789
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