  We prove a long standing conjecture concerning the fencing problem in the
plane: among planar convex sets of given area, prove that the disc, and only
the disc maximizes the length of the shortest area-bisecting curve. Although it
may look intuitive, the result is by no means trivial since we also prove that
among planar convex sets of given area the set which maximizes the length of
the shortest bisecting chords is the so-called Auerbach triangle.
