# Spherical Discrepancy Minimization and Algorithmic Lower Bounds for Covering the Sphere

## TL;DR

This paper introduces the spherical discrepancy problem, proves its computational hardness, develops an optimal algorithm, and establishes new lower bounds for sphere covering using Gaussian space techniques.

## Contribution

It defines the spherical discrepancy problem, proves its APX-hardness, and provides an algorithm with optimal error bounds, along with new lower bounds for sphere covering.

## Key findings

- Spherical discrepancy is APX-hard.
- Developed a multiplicative weights algorithm with optimal bounds.
- Established new lower bounds for covering the sphere with caps.

## Abstract

Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x, v_i\rangle$. We show that \textsc{Spherical Discrepancy} is APX-hard and develop a multiplicative weights-based algorithm that achieves optimal worst-case error bounds up to lower order terms. We use our algorithm to give the first non-trivial lower bounds for the problem of covering a hypersphere by hyperspherical caps of uniform volume at least $2^{-o(\sqrt{n})}$. We accomplish this by proving a related covering bound in Gaussian space and showing that in this \textit{large cap regime} the bound transfers to spherical space. Up to a log factor, our lower bounds match known upper bounds in the large cap regime.