Isoperimetric property of balls for the distributional fractional s-perimeter

Establish whether balls minimize the distributional fractional \(s\)-perimeter among measurable sets of prescribed positive finite volume; specifically, prove or disprove that every measurable \(E\subset\mathbb{R}^N\) with \(0<|E|<\infty\) satisfies \(\frac{|D^s\mathbf{1}_{B_1}|(\mathbb{R}^N)}{|B_1|^{(N-s)/N}}\leq\frac{|D^s\mathbf{1}_E|(\mathbb{R}^N)}{|E|^{(N-s)/N}}\).

Background

The distributional fractional $s$-perimeter is introduced through the fractional variation of characteristic functions. The paper explains that a Pólya–Szegő inequality at $p=1$ would imply the sharp isoperimetric inequality asserting that balls minimize the normalized distributional fractional $s$-perimeter.

Although a non-sharp isoperimetric inequality is available, the sharp minimizing property of balls remains unresolved. The paper’s counterexamples to the Pólya–Szegő inequality do not settle this geometric question.