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

Any finite, separately convex, positively homogeneous function on $\mathbb{R}^2$ is convex. This was first established by the first author ["Direct methods in calculus of variations", Springer-Verlag (1989)]. Here 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 {\it perspective} of a convex function $f$, namely, the function $(x,y)\to yf(x/y)$, $y>0$. In recent works of the second author [Math. Programming 89A (2001) 505--516; J. Optimization Theory Appl. 126 (2005) 175--189 and 357--366], the perspective has been substantially generalized by considering functions of the form $(x,y) \to g(y)f(x/g(y))$, with suitable assumptions on $g$. Here, this {\it generalized perspective} is shown to be a powerful tool for the analysis of convexity properties of parametrized families of matrix functions.