Geometric systems of unbiased representatives

Let P be a finite point set in R^d, B be a bicoloring of P and O be a family of geometric objects (that is, intervals, boxes, balls, etc). An object from O is called balanced with respect to B if it contains the same number of points from each color of B. For a collection B of bicolorings of P, a geometric system of unbiased representatives (G-SUR) is a subset O^′⊆O such that for any bicoloring B of B there is an object in O^′ that is balanced with respect to B. We pose and study problems on finding G-SURs. We obtain general bounds on the size of G-SURs consisting of intervals, size-restricted intervals, axis-parallel boxes and Euclidean balls. We show that the G-SUR problem is NP-Hard even in the simple case of points on a line and interval ranges. Furthermore, we study a related problem on determining the size of the largest and smallest balanced intervals for points on the real line with a random distribution and coloring. Our results are a natural extension to a geometric context of the work initiated by Balachandran et al. (Discrete Mathematics, 2018) on arbitrary systems of unbiased representatives.