Geometric discrepancy theory

Geometric discrepancy theory is a sub-field of discrepancy theory, that deals with balancing geometric sets, such as intervals or rectangles. The general research question in this field is: given a set of points in a geometric space, and a set of objects in the same space, can we color each point in one of two different colors (e.g. black and white), such that each object contains roughly the same number of points of each color?

Formally, the discrepancy of an object is defined as the difference between the number of white points and the number of black points in that object; the objective is to color the points such that the maximum discrepancy of an object is as small as possible.

Intervals
In the simplest geometric discrepancy setting, the set of objects is the set of all sub-intervals of the real interval [0,1]. In this setting, it is possible to attain discrepancy 1: simply color the points alternately black - white - black - white - etc. Then, the discrepancy of every interval is either 0 or 1.

The problem becomes more challenging when the points are not available in advance, but arrive one by one, and each point should be colored immediately when it arrives. This setting is called the "Online Interval Discrepancy". Jiang, Kulkarni and Singla prove that:


 * No online algorithm can guarantee a constant discrepancy.
 * Randomly coloring each point when it arrives gives $$\tilde{O}(\sqrt{n})$$ expected discrepancy.
 * If the point arrival is adversarial, the discrepancy of any online algorithm is $$\Omega(\sqrt{n})$$.
 * If the point arrival is stochastic, there is an efficient algorithm that guarantees $$O(n^{c/\log{\log{n}}})$$ discrepancy, for some universal constant c, with high probability (i.e. with probability 1-1/poly(n), where the exponent of the polynomial depends on c).

Rectangles
Tusnadi asked what is the discrepancy when the set of objects is the set of axes-parallel rectangles contained in the unit square.

When the set of objects is the set of all rectangles (possibly rotated), then:
 * Beck proved that the discrepancy is at least Ω(log n) and at most O(log4n).
 * Nikolov proved that the discrepancy is at most O(log1.5 n).


 * Beck proved that the discrepancy is at least Ω(n1/4-ε) and at most O(n1/2+ε) for any ε>0.

Stripes
When the set of objects is the set of stripes -- rectangles of the form [a,b]x[0,1] and [0,1]x[a,b], the setting is equivalent to the problem of "two permutations": given two permutations on a set of n elements, we should color each element either black or white, such that the discrepancy in each interval of each permutation is minimized.


 * Spencer proved that it is possible to attain a discrepancy of at most 2.

Convex polytopes
Nikolov studied a more general setting, where the set of objects is induced by dilations and translations of a fixed convex polytope. He proved upper and lower bounds on the discrepancy.