The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
Algebraic optimization: the Fermat-Weber location problem
Mathematical Programming: Series A and B
Geometric discrepancy theory and uniform distribution
Handbook of discrete and computational geometry
Fast approximations for sums of distances, clustering and the Fermat--Weber problem
Computational Geometry: Theory and Applications
New Bounds on the Average Distance from the Fermat-Weber Center of a Planar Convex Body
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
New bounds on the average distance from the Fermat-Weber center of a planar convex body
Discrete Optimization
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A Steiner star for a set P of n points in R^d connects an arbitrary point in R^d to all points of P, while a star connects one of the points in P to the remaining n-1 points of P. All connections are realized by straight line segments. Let the length of a graph be the total Euclidean length of its edges. Fekete and Meijer showed that the minimum star is at most 2 times longer than the minimum Steiner star for any finite point configuration in R^d. The supremum of the ratio between the two lengths, over all finite point configurations in R^d, is called the star Steiner ratio in R^d. It is conjectured that this ratio is 4/@p=1.2732... in the plane and 4/3=1.3333... in three dimensions. Here we give upper bounds of 1.3631 in the plane, and 1.3833 in 3-space. These estimates yield improved upper bounds on the maximum ratios between the minimum star and the maximum matching in two and three dimensions. We also verify that the conjectured bound 4/@p in the plane holds in two special cases. Our method exploits the connection with the classical problem of estimating the maximum sum of pairwise distances among n points on the unit sphere, first studied by Laszlo Fejes Toth. It is quite general and yields the first nontrivial estimates below 2 on the star Steiner ratios in arbitrary dimensions. We show, however, that the star Steiner ratio in R^d tends to 2 as d goes to infinity. As it turns out, our estimates are related to the classical infinite Wallis product: @p2=@?"n"="1^~(4n^24n^2-1)=21@?23@?43@?45@?65@?67@?87@?89....