The hardest halfspace

We study the approximation of halfspaces h:{0,1}^n→{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^-Ω(n) , between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of log n versus Ω(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 Ω(n/4^n) for communication with unbounded versus weakly unbounded error.