  We introduce a notion of convexity with respect to a one-dimensional operator
and with this notion find a one-parameter family of different convexities that
interpolates between classical convexity and quasiconvexity. We show that, for
this interpolation family, the convex envelope of a continuous boundary datum
in a strictly convex domain is continuous up to the boundary and is
characterized as being the unique viscosity solution to the Dirichlet problem
in the domain for a certain fully nonlinear partial differential equation that
involves the associated operator. In addition we prove that the convex
envelopes of a boundary datum constitute a one-parameter curve of functions
that goes from the quasiconvex envelope to the convex envelope being continuous
with respect to uniform convergence. Finally, we also show some regularity
results for the convex envelopes proving that there is an analogous to a
supporting hyperplane at every point and that convex envelopes are $C^1$ if the
boundary data satisfies in particular $NV$-condition we introduce.
