Notes on the trace problem for separately convex functions

TL;DR

This paper investigates the properties of functions that are convex along coordinate axes and examines the characteristics of their diagonal traces, providing conditions and examples in two dimensions.

Contribution

It offers necessary and sufficient conditions for the trace of separately convex functions in two dimensions and explores the case for concave functions.

Findings

Characterization of traces for convex functions in 2D

Examples illustrating limitations of current approach

Complete characterization for concave functions in 2D

Abstract

We discuss the following question: For a function f of two or more variables which is convex in the directions of coordinate axes, how can its trace g(x) = f(x, x, ..., x) look like? In the two-dimensional case, we provide some necessary and sufficient conditions, as well as some examples illustrating that our approach does not seem to be appropriate for finding a characterization in full generality. For a concave function g, however, a characterization in the two-dimensional case is established.