On the best constant in the nonlocal isoperimetric inequality of Almgren and Lieb

Abstract

In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order $W^{s,p}$. The case $p = 2$ of their result implies the nonlocal isoperimetric inequality \[ \frac{P_s(E)}{|E|^{\frac{N-2s}N}} \ge \frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}},\ \ \ \ \ \ \ 0<s<1/2, \] where $P_s$ indicates the fractional $s$-perimeter, and $B_1$ is the unit ball in $\RN$. In this note we explicitly compute the best constant, and show that for any $0<s<1/2$, one has \[ \frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}} = \frac{N \pi^{\frac N2 + s} \G(1-2s)}{s \G(\frac N2+1)^{\frac{2s}N} \G(1-s)\G(\frac{N+2-2s}{2})}. \]