Lp approximation by subsets of convex functions of several variables

Abstract

If S is a bounded convex subset of R m , the problem is to find a best approximation to a function in L p ( S), 1 ⩽ p ⩽ ∞, by an arbitrary subset of convex functions. An existence theorem for a best approximation is established under a certain condition on the subset. In particular, a best convex approximation exists. Also investigated are properties of norm-bounded subsets and L p -convergent sequences of convex functions.