  We consider the domain dependence of the best constant in the subcritical
fractional Sobolev constant,
  $$
  \lambda_{s,p}(\Omega):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in
C^\infty_c(\Omega),\,\, \|u\|_{L^p(\Omega)}=1 \right\},
  $$ where $s\in (0,1)$, $\Omega$ is bounded of class $C^{1,1}$ and $p\in [1,
\frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we
derive formula for the one-sided shape derivative of the mapping $\Omega\mapsto
\lambda_{s,p}(\Omega)$ under domain perturbations. In the case where $
\lambda_{s,p}(\Omega)$ admits a unique positive minimizer (e.g. $p=1$ or
$p=2$), our result implies a nonlocal version of the classical variational
Hadamard formula for the first eigenvalue of the Dirichlet Laplacian on
$\Omega$. Thanks to the formula for our one-sided shape derivative, we
characterize smooth local minimizers of $\lambda_{s,p}(\Omega)$ under
volume-preserving deformations, and we find that they are balls if $p\in
\{1\}\cup [2,\infty)$. Finally, we consider the maximization problem for
$\lambda_{s,p}(\Omega)$ among annular-shaped domains of fixed volume of the
type $B\setminus \overline B'$, where $B$ is a fixed ball and $B'$ is ball
whose position is varied within $B$. We prove that, for $p\in \{1,2\}$, the
value $\lambda_{s,p}(B\setminus \overline B')$ is maximal when the two balls
are concentric.
