Approximating the ball by a Minkowski sum of segments with equal length
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 \textit{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)].
scientific article; zbMATH DE number 168597