A proof of the Oja depth conjecture in the plane

Authors:
Nabil H. Mustafa;Hans Raj Tiwary;Daniel Werner
Affiliations:
Laboratoire d'Informatique Gaspard-Monge, Université Paris-Est, Equipe A3SI, ESIEE, Paris, France;Department of Mathematics, Université Libre de Bruxelles, Belgium;Institut für Informatik, FU Berlin, Germany
Venue:
Computational Geometry: Theory and Applications
Year:
2014

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Abstract

Given a set P of n points in the plane, the Oja depth of a point x@?R^2 is defined to be the sum of the areas of all triangles defined by x and two points from P, normalized with respect to the area of the convex hull of P. The Oja depth of P is the minimum Oja depth of any point in R^2. The Oja depth conjecture states that any set P of n points in the plane has Oja depth at most n^2/9. This bound would be tight as there are examples where it is not possible to do better. We present a proof of this conjecture. We also improve the previously best bounds for all R^d, d=3, via a different, more combinatorial technique.