# The Hardest Halfspace

## TL;DR

This paper constructs the hardest possible halfspace for polynomial and rational approximation, and applies this to achieve nearly optimal separations in communication complexity, advancing understanding of computational hardness.

## Contribution

It explicitly constructs the hardest halfspace for approximation, matching trivial bounds, and uses this to improve communication complexity separations in multiple models.

## Key findings

- Established polynomial and rational approximation lower bounds for the hardest halfspace.
- Achieved near-optimal separation between sign-rank and discrepancy in communication complexity.
- Extended results to the $k$-party number-on-the-forehead model, improving previous bounds.

## Abstract

We study the approximation of halfspaces $h:\{0,1\}^n\to\{0,1\}$ in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the "hardest" halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all halfspaces. This completes a lengthy line of work started by Myhill and Kautz (1961). As an application, we construct a communication problem that achieves essentially the largest possible separation, of $O(n)$ versus $2^{-\Omega(n)},$ between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of $\log n$ versus $\Omega(n)$ between the communication complexity with unbounded versus weakly unbounded error, improving quadratically on previous constructions and completing a line of work started by Babai, Frankl, and Simon (FOCS 1986). Our results further generalize to the $k$-party number-on-the-forehead model, where we obtain an explicit separation of $\log n$ versus $\Omega(n/4^{n})$ for communication with unbounded versus weakly unbounded error. This gap is a quadratic improvement on previous work and matches the state of the art for number-on-the-forehead lower bounds.