  We study the sharp constant in the Hardy inequality for fractional Sobolev
spaces defined on open subsets of the Euclidean space. We first list some
properties of such a constant, as well as of the associated variational
problem. We then restrict the discussion to open convex sets and compute such a
sharp constant, by constructing suitable supersolutions by means of the
distance function. Such a method of proof works only for $s\,p\ge 1$ or for
$\Omega$ being a half-space. We exhibit a simple example suggesting that this
method can not work for $s\,p<1$ and $\Omega$ different from a half-space. The
case $s\,p<1$ for a generic convex set is left as an interesting open problem,
except in the Hilbertian setting (i.e. for $p=2$): in this case we can compute
the sharp constant in the whole range $0<s<1$. This completes a result which
was left open in the literature.
