An alternative approach to Shnirelman's inequality

In this paper we examine the discrete Shnirelman's inequality [Shnirelman A., 1985], which relates the $L^2$-distance of two discrete configurations of a fluid to the $L^1_tL^2_x$-norm of the vector field connecting them. Our proof is inspired by [Shnirelman A., 1985], where it was obtained $\alpha=\frac{1}{64}$ in dimension $\nu=2$, while here we get $\alpha\geq\frac{2}{7}$. Moreover we prove that $\alpha\geq\frac{1}{\nu+1}$ for any dimension $\nu\geq 3$. We point out that, even if this does not improve the bound in the continuous version, where it was proved that $\alpha\geq\frac{2}{4+\nu}$, with $\nu\geq 3$, our bound is the best one achieved for the $2$-dimensional case. Our method uses an alternative approach based on volume estimates of permutations, which count the number of maximum cubes that are moved by a permutation $P$.