Efficient Algorithms for Discrepancy Minimization in Convex Sets

Understand

A result of Spencer states that every collection of $n$ sets over a universe of size $n$ has a coloring of the ground set with $\{-1,+1\}$ of discrepancy $O(\sqrt{n})$.

• A geometric generalization of this result was given by Gluskin (see also Giannopoulos) who showed that every symmetric convex body $K\subseteq R^n$ with Gaussian measure at least $e^{-\epsilon n}$, for a small $\epsilon>0$, contains a point $y\in K$ where a constant fraction of coordinates of $y$ are in $\{-1,1\}$.
• This is often called a partial coloring result.
• While both these results were inherently non-algorithmic, recently Bansal (see also Lovett-Meka) gave a polynomial time algorithm for Spencer's setting and Rothvo\ss gave a randomized polynomial time algorithm obtaining the same guarantee as the result of Gluskin and Giannopoulos.