k-Centerpoints Conjectures for Pointsets in ℝd

In this paper, we introduce the notion of [Formula: see text]-centerpoints for any set [Formula: see text] of [Formula: see text] points in [Formula: see text]. These unify and generalize previous results such as the classical centerpoint theorem,1 and the recently-proven ray-shooting theorem.2 We define two variants: affine [Formula: see text]-centerpoints, and topological [Formula: see text]-centerpoints. We prove their equivalence in [Formula: see text], and conjecture that these are in fact equivalent in any dimension. We present the first non-trivial bounds for these problems in [Formula: see text], as well as present several conjectures related to them.