  The projection onto the epigraph or a level set of a closed proper convex
function can be achieved by finding a root of a scalar equation that involves
the proximal operator as a function of the proximal parameter. This paper
develops the variational analysis of this scalar equation. The approach is
based on a study of the variational-analytic properties of general convex
optimization problems that are (partial) infimal projections of the the sum of
the function in question and the perspective map of a convex kernel. When the
kernel is the Euclidean norm squared, the solution map corresponds to the
proximal map, and thus the variational properties derived for the general case
apply to the proximal case. Properties of the value function and the
corresponding solution map -- including local Lipschitz continuity, directional
differentiability, and semismoothness -- are derived. An SC$^1$ optimization
framework for computing epigraphical and level-set projections is thus
established. Numerical experiments on 1-norm projection illustrate the
effectiveness of the approach as compared with specialized algorithms
