Upper bounds for the constants of Bennett's inequality and the Gale–Berlekamp switching game

Abstract

AbstractIn 1977, G. Bennett proved, by means of nondeterministic methods, an inequality that plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for and all positive integers , there exists a bilinear form with coefficients ±1 satisfying for a certain constant depending just on ; moreover, the exponents of cannot be improved. In this paper, using a constructive approach, we prove that whenever or ; our techniques are applied to provide new upper bounds for the constants of the Gale–Berlekamp switching game, improving estimates obtained by Brown and Spencer in 1971 and by Carlson and Stolarski in 2004.