Journal of Convex Analysis 15 (2008), No. 2, 271--284
Copyright Heldermann Verlag 2008
The Role of Perspective Functions in Convexity, Polyconvexity, Rank-One Convexity and Separate Convexity
Bernard Dacorogna
Section de Mathématiques, Ecole Polytechnique Fédérale, 1015 Lausanne, Switzerland
bernard.dacorogna@epfl.ch
Pierre Maréchal
Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse 4, France
marechal@mip.ups-tlse.fr
[Abstract-pdf]
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.
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