An alternative approach to the discrete Shnirelman’s inequality
Year: 2025
Type: article
Abstract: In this paper we examine the discrete Shnirelman’s inequality (Shnirelman 1985 Math. USSR Sb. 56 82–109), which relates the L 2-distance of two discrete configurations of a fluid to the L t 1 L x 2 -norm of the vector field connecting them. Our proof is inspired by (Shnirelman 1985 Math. USSR Sb. 56 82–109), where it was obtained α = 1 64 in dimension ν = 2, while here we get α ⩾ 2 7 . Moreover, this bound can be taken α ⩾ 1 ν + 1 for any dimension ν ⩾ 3 . This is the best bound achieved in a purely discrete context, and connections to the continuous Shnirelman’s inequality are discussed. 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.
Source: Nonlinearity
Author: Martina Zizza
Institution: Max Planck Institute for Mathematics in the Sciences
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,325
OpenAlex ID: W4407785922