The role of perspective functions in convexity, polyconvexity, rank-one convexity and separate convexity

Any finite, separately convex, positively homogeneous function on R is convex. This was first established in [1]. In this paper, we give a new and concise proof of this result, and we show that it fails in higher dimension. The key of the new proof is the notion of perspective of a convex function f , namely, the function (x, y) → yf(x/y), y > 0. In recent works [9, 10, 11], the perspective has been substantially generalized by considering functions of the form (x, y) → g(y)f(x/g(y)), with suitable assumptions on g. Here, this generalized perspective is shown to be a powerful tool for the analysis of convexity properties of parametrized families of matrix functions.