Suppose E is a compact subset of R^n, and we are given a function f, mapping E to the real numbers. How can we tell if the function lies on a smooth convex function? Can we construct an almost optimal, smooth, convex interpolant of the function? These are examples of Whitney-type extension and trace problems; while theoretical, they are driven by practical questions of interpolation of data, where convexity is a natural constraint. I will begin with an answer to these questions by presenting work of mine proving there is a Finiteness Principle for the non-linear space of strongly convex functions in $C^{1,1}(\mathbb{R}^n)$. This work is the first attempt to understand the constrained interpolation problem for convex functions in $C^{1,1}(\mathbb{R}^n)$. We will spend time discussing useful techniques and tools from analysis as well as important open problems in the field.