Efficient Algorithms for Discrepancy Minimization in Convex Sets

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. This paper has several related results. First we prove another constructive version of the result of Gluskin and Giannopoulos via an optimization of a linear function. This implies a linear programming based algorithm for combinatorial discrepancy obtaining the same result as Spencer. Our second result gives a new approach to obtains partial colorings and shows that every convex body $K\subseteq R^n$, possibly non-symmetric, 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 $\{-1,1\}$. Finally, we give a simple proof that shows that for any $\delta >0$ there exists a constant $c>0$ such that given a body $K$ with $\gamma_n(K)\geq \delta$, a uniformly random $x$ from $\{-1,1\}^n$ is in $cK$ with constant probability. This gives an algorithmic version of a special case of the result of Banaszczyk.