K-Centerpoints Conjectures for Pointsets in R^d

International audienceIn this paper, we introduce the notion of k-centerpoints for any set P of n points in R^d. These unify and generalize previous results such as the classical centerpoint theorem, and the recently-proven ray-shooting theorem. We define two variants: affine k-centerpoints, and topological k-centerpoints. We prove their equivalence in R^2 , and conjecture that these are in fact equivalent in any dimension