Centerpoints and Tverberg's technique

Using a technique that Tverberg and Vrecica (1993) [16] discovered to give a surprisingly simple proof of Tverberg?s theorem, we show the following extension of the centerpoint theorem. Given any set P of n points in the plane, and a parameter 1/3 ≤ c ≤ 1, one can always find a disk D such that any closed half-space containing D contains at least cn points of P. Furthermore, D contains at most (3c ? 1)n/2 points of P (the case c = 1 is trivial ? take any D containing P; the case c = 1/3 is the centerpoint theorem). We also show that, for all c, this bound is tight up to a constant factor. We extend the upper bound to R^d. Specifically, we show that given any set P of n points, one can find a ball D containing at most ((d + 1)c ? 1)n/d points of P such that any half-space containing D contains at least cn points of P.