The hardest halfspace

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