A proof of the Oja depth conjecture in the plane

Given a set P of n points in the plane, the Oja depth of a point x∈R2 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 R2. The Oja depth conjecture states that any set P of n points in the plane has Oja depth at most n2/9. 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 Rd, d⩾3, via a different, more combinatorial technique.