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

In 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 p1,p2∈[1,∞]$p_{1},p_{2} \in [1,\infty ]$ and all positive integers n1,n2$n_{1},n_{2}$, there exists a bilinear form An1,n2:(Rn1,∥·∥p1)×(Rn2,∥·∥p2)⟶R$A_{n_{1},n_{2}}\colon (\mathbb {R}^{n_{1}},\Vert \cdot \Vert _{p_{1}}) \times (\mathbb {R}^{n_{2}},\Vert \cdot \Vert _{p_{2}}) \longrightarrow \mathbb {R}$ with coefficients ±1 satisfying An1,n2⩽Cp1,p2maxn11−1p1n2max12−1p2,0,n21−1p2n1max12−1p1,0$$\begin{eqnarray*} &&\hspace*{13pc}{\left\Vert A_{n_{1},n_{2}}\right\Vert} \leqslant C_{p_{1},p_{2}}\max {\left\lbrace n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max {\left\lbrace \frac{1}{2}-\frac{1}{p_{2} },0\right\rbrace} },\right.}\\ &&\hspace*{21pc}{\left.n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max {\left\lbrace \frac{1}{2} -\frac{1}{p_{1}},0\right\rbrace} }\right\rbrace} \end{eqnarray*}$$for a certain constant Cp1,p2$C_{p_{1},p_{2}}$ depending just on p1,p2$p_{1},p_{2}$; moreover, the exponents of n1,n2$n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach, we prove that Cp1,p2⩽8/5$C_{p_{1},p_{2}}\leqslant \sqrt {8/5}$ whenever p1,p2∈[2,∞]$p_{1},p_{2}\in [ 2,\infty ]$ or p1=p2=p∈[1,∞]$p_{1}=p_{2}=p\in [ 1,\infty ]$; 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. [ABSTRACT FROM AUTHOR]