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 by the area of 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 would be optimal 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 tech-nique.