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Constructive approximation of a ball by polytopes. (English) Zbl 0799.52005
Let \(S_ n(r)\) denote the \(n\)-dimensional ball of radius \(r\) with centre at the origin. The author gives an explicit construction of \(m\) unit vectors \(v_ i \in S_ n(1)\) such that the convex hull \(\text{conv} \{v_ 1, \dots, v_ m\}\) contains a ball \(S_ n(r)\) with \(r=c \sqrt {n^{-1} \log (m/n)}\) for \(2n \leq m \leq k^ n\) with any constant \(k>1\). The constant \(c\) is independent on the dimension \(n\), and the value \(r\) is asymptotically optimal \((n \to \infty)\) but only for \(2n \leq m\). Contrary to the title the author does not consider the (best) approximating polytopes (with minimal number of vertices) of the ball in the sense of the minimal Hausdorff distance, for example.
Reviewer: E.Hertel (Jena)

MSC:
52A27 Approximation by convex sets
52B55 Computational aspects related to convexity
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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References:
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