  Let $P$ be a set of points in $\mathbb{R}^d$, $B$ a bicoloring of $P$ and
$\Oo$ a family of geometric objects (that is, intervals, boxes, balls, etc). An
object from $\Oo$ 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 $\Oo'\subseteq\Oo$ such that for any bicoloring $B$ of $\B$ there is an
object in $\Oo'$ that is balanced with respect to $B$.
  We study the problem of 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. on arbitrary systems of unbiased
representatives.
