  We consider an optimal rearrangement minimization problem involving the
fractional Laplace operator $(-\Delta)^s$, $0<s<1$, and Gagliardo-Nirenberg
seminorm $|u|_s$. We prove the existence of the unique minimizer, analyze its
properties as well as derive the non-local and highly non-linear PDE it
satisfies $$ -(-\Delta)^s U-\chi_{\{U\leq 0\}}\min\{-(-\Delta)^s
U^+;1\}=\chi_{\{U>0\}}, $$ which happens to be the fractional analogue of the
normalized obstacle problem $\Delta u=\chi_{\{u>0\}}$.
  A new section analyzing $s \to 1$ has been added.
