  Recently, Simons provided a lemma for a support function of a closed convex
set in a general Hilbert space and used it to prove the geometry conjecture on
cycles of projections. In this paper, we extend Simons's lemma to closed convex
functions, show its connections to Attouch-Thera duality, and use it to
characterize (classical and phantom) cycles and gap vectors of proximal
mappings.
