A note on spherical discrepancy

TL;DR

This paper proves a generalized, non-algorithmic version of a discrepancy result using convex geometry, extending the Komlós conjecture to spherical discrepancy.

Contribution

It introduces a simple convex geometry-based proof for a relaxed form of the Komlós conjecture involving spherical discrepancy.

Findings

Spherical discrepancy version of the Komlós conjecture is true.

A short, convex geometry-based proof is provided.

The result extends prior boolean discrepancy results.

Abstract

A non-algorithmic, generalized version of a recent result, asserting that a natural relaxation of the Koml\'os conjecture from boolean discrepancy to spherical discrepancy is true, is proved by a very short argument using convex geometry.