Rigidity for measurable sets (Q2685681)

We define the \(r\)-\textit{critical} sets as the sets of \(\Omega\subset\mathbb R^d\) with finite Lebesgue measure such that the Lebesgue measure of \(\Omega\cap B_r(x)\) is a constant when \(x\) varies in \(\partial^*\Omega\), the essential boundary of \(\Omega\), where \(B_r(x)\) is the standard Euclidean ball with center \(x\) and of radius \(r\). We say that a measurable set \(\Omega\) in \(\mathbb R^d\) is \(r\)-\textit{degenerate} if \[ \inf_{(x_1,x_2)\in\partial^*\Omega\times\partial^*\Omega}\left[\frac{|\Omega\cap(B_r(x_1)\Delta B_r(x_2))|} {\|x_1-x_2\|}\right] =0. \] The authors assume that \(\Omega\) is \(r\)-\textit{critical} and not \(r\)-\textit{degenerate} and show that \(\Omega\) is equivalent to a finite union of balls of the same radius \(R>\frac{r}{2}\), at mutual distance larger than or equal to \(r\).