  We study how the supporting hyperplanes produced by the projection process
can complement the method of alternating projections and its variants for the
convex set intersection problem. For the problem of finding the closest point
in the intersection of closed convex sets, we propose an algorithm that, like
Dykstra's algorithm, converges strongly in a Hilbert space. Moreover, this
algorithm converges in finitely many iterations when the closed convex sets are
cones in $\mathbb{R}^{n}$ satisfying an alignment condition. Next, we propose
modifications of the alternating projection algorithm, and prove its
convergence. The algorithm converges superlinearly in $\mathbb{R}^{n}$ under
some nice conditions. Under a conical condition, the convergence can be finite.
Lastly, we discuss the case where the intersection of the sets is empty.
