  This paper introduces in a natural way a notion of horizontal convex
envelopes of continuous functions in the Heisenberg group. We provide a
convexification process to find the envelope in a constructive manner. We also
apply the convexification process to show h-convexity of viscosity solutions to
a class of fully nonlinear elliptic equations in the Heisenberg group
satisfying a certain symmetry condition. Our examples show that in general one
cannot expect h-convexity of solutions without the symmetry condition.
