Geometric discrepancy theory[1] 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?
In this setting, it is possible to attain discrepancy 1: simply color the points alternately black - white - black - white - etc.
Jiang, Kulkarni and Singla prove that:[2]: Sec.3.2 Their proof uses a reduction to the problem of Online Tree Balancing, which is a problem of discrepancy in which the set of objects is the set of sub-trees of a complete m-ary tree with height h. For this problem, they prove that, if
for a sufficiently large constant C, and m ≥ 100, then there is an online algorithm that attains discrepancy
Improving previous results by Roth, Schmidt, Beck, Bohus, and Srinivasan, he proved an upper bound of
Jiang, Kulkarni and Singla[2] study the online setting with stochastic point arrival, and prove that: Matousek[5] and Nikolov[4] studied a more general setting, where the set of objects is induced by dilations and translations of a fixed convex polytope.