Sliding a convex set under a convex function

Abstract

An existing form of approximating a non-differentiable function, f, is to slide a convex set, A, along the graph of f and take the lower boundary to form a differentiable function g. Our work reveals that sliding a convex set, A, under the graph of f gives the same approximation function, g, as sliding the set A-A along f. Additionally, sliding a convex set, A, under f is the same as sliding -A under f, or A-A along f. Due to this, if A is smooth from above or below, the approximation function g is essentially differentiable