\(L_ p\) approximation by subsets of convex functions of several variables (Q1813614)

Let \(S\) be a bounded convex subset of \(\mathbb{R}^ m\). The author finds a best approximation to a function in the \(L_ p(S)\), \(1\leq p<\infty\), by an arbitrary subset of convex functions. He establishes three theorems on the properties of norm-bounded subsets and convergent sequences of convex functions as well as on the existence of a best approximation.