We use quadrature formulas with equal weights in order to constructN point sets on spheres ind-space (d ≥ 3) which are almost optimal with respect to a discrepancy concept, based on distance functions (potentials) and distance functionals (energies). By combining this approach with the probabilistic method, we obtain almost best possible approximations of balls by zonotopes, generated byN segments of equal length.
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Editors' note: We learned with sadness of Gerold Wagner's untimely death as a result of an avalanche in the Alps shortly after the submission of this paper. When one of the referees, Joram Lindenstrauss, suggested that Wagner's results might be extended to dimensions >6, we invited Professor Lindenstrauss to submit a paper containing that extension which we would publish alongside the Wagner paper. The result is the paper by Bourgain and Lindenstrauss that follows the present one.
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Wagner, G. On a new method for constructing good point sets on spheres. Discrete Comput Geom 9, 111–129 (1993). https://doi.org/10.1007/BF02189312
- Convex Body
- Random Point
- Discrete Comput Geom
- Quadrature Formula
- Main Lemma