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Approximating the ball by a Minkowski sum of segments with equal length. (English) Zbl 0770.52005
The following theorem is proved. For every \(n\in\mathbb{N}\), there is a constant \(c(n)\) so that the \(n\)-dimensional Euclidean ball can be approximated up to \(\varepsilon>0\) in the Hausdorff metric by a sum of \[ N\leq c(n)(\varepsilon^{-2}|\log\varepsilon|)^{(n- 1)/(n+2)} \] segments of equal length. For \(n\leq 6\), this was proved by G. Wagner [Discrete Comput. Geom. 9, 111-129 (1993)]. The authors use a different method, in part also related to numerical integration with equal weights. The same result, but with sums of segments of not necessarily equal lengths, had been obtained earlier by the same authors [Isr. J. Math. 64, No. 1, 25-31 (1988; Zbl 0667.52001)].

MSC:
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52A27 Approximation by convex sets
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References:
[1] Bourgain, J.; Lindenstrauss, J., Distribution of points on spheres and approximation by zonotopes, Israel J. Math., 64, 25-31, (1988) · Zbl 0667.52001
[2] Bourgain, J.; Lindenstrauss, J.; Milman, V., Approximation of zonoids by zonotopes, Acta Math., 162, 73-141, (1989) · Zbl 0682.46008
[3] Betke, U.; McMullen, P., Estimating the sizes of convex bodies by projections, J. London Math. Soc., 27, 525-538, (1983) · Zbl 0487.52005
[4] G. Wagner, On a new method for constructing good point sets on spheres,Discrete Comput. Geom., this issue, pp. 111-129.
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