An alternative approach to the discrete Shnirelman’s inequality

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> -norm of the vector field connecting them. Our proof is inspired by (Shnirelman 1985 Math. USSR Sb. 56 82–109), where it was obtained <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>64</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> in dimension ν = 2, while here we get <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>7</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> . Moreover, this bound can be taken <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> for any dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . 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 .