Lower bounds for weak epsilon-nets and stair-convexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
Improving the first selection lemma in R3
Proceedings of the twenty-sixth annual symposium on Computational geometry
Overlap properties of geometric expanders: extended abstract
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
A proof of the Oja depth conjecture in the plane
Computational Geometry: Theory and Applications
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The following result was proved by Bárány in 1982: For every d≥1, there exists c d0 such that for every n-point set S in ℝd , there is a point p∈ℝd contained in at least c d n d+1−O(n d ) of the d-dimensional simplices spanned by S. We investigate the largest possible value of c d. It was known that c d≤1/(2d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c d≤(d+1)−(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is c d≥γ d:=(d 2+1)/((d+1)!(d+1)d+1); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γ d n d+1+O(n d ) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved. We also prove that for every n-point set S⊂ℝd , there exists a (d−2)-flat that stabs at least c d,d−2 n 3−O(n 2) of the triangles spanned by S, with $c_{d,d-2}\ge\frac{1}{24}(1-1/(2d-1)^{2})$. To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in ℝd can be divided into 4d−2 equal parts by 2d−1 hyperplanes intersecting in a common (d−2)-flat.