Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Thursday, September 26, 2019, 12:15 pm

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Daniel Bertschinger

Given any point set P ⊆ ℝ^{d} consisting of n points, a weighted ε-net is defined as a set of points p_{1},…,p_{k} and some values ε = (ε_{1},…,ε_{k}) such that every set in the range space containing more than ε_{i}n points of P contains at least i of the points p_{1},…,p_{k}. It is a notion of approximation for point sets in ℝ^{d} similar to ε-nets and ε-approximations, where it is stronger than the former and weaker than the latter. We analyze small weak weighted ε-nets with respect to convex sets and show that for any point set of size n in ℝ^{d} and for any positive α ≤ β with (i) dα + β ≥ d and (ii) α ≥ (2d-1)/(2d+1) we can find two points p_{1} and p_{2} such that each convex set containing more than αn points of P contains at least one of p_{1} and p_{2} and each convex set containing more than βn points contains both p_{1} and p_{2}. For axis-parallel boxes in ℝ^{d} instead of convex sets we can improve the conditions.

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