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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

Pierre Maréchal
Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse 4, France


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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