  We give a partial negative answer to a question left open in a previous work
by Brasco and the first and third-named authors concerning the sharp constant
in the fractional Hardy inequality on convex sets. Our approach has a
geometrical flavor and equivalently reformulates the sharp constant in the
limit case $p=1$ as the Cheeger constant for the fractional perimeter and the
Lebesgue measure with a suitable weight. As a by-product, we obtain new lower
bounds on the sharp constant in the $1$-dimensional case, even for non-convex
sets, some of which optimal in the case $p=1$.
