An alternative approach to Shnirelman's inequality

In this paper we examine the discrete Shnirelman's inequality [Shnirelman A., 1985], which relates the L²-distance of two discrete configurations of a fluid to the L¹ₜL²ₓ-norm of the vector field connecting them. Our proof is inspired by [Shnirelman A., 1985], where it was obtained α=1/64 in dimension ν=2, while here we get α≥2/7. Moreover we prove that α≥1/ν+1 for any dimension ν≥ 3. We point out that, even if this does not improve the bound in the continuous version, where it was proved that α≥2/4+ν, with ν≥ 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.