The Role of Perspective Functions in Convexity, Polyconvexity, Rank-One Convexity and Separate Convexity
Bernard Dacorogna, Pierre Maréchal · Journal of convex analysis · 2008
Any finite, separately convex, positively homogeneous function on \mathbb{R}^2 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 perspective of a convex function f f , namely, the function (x,y)\to yf(x/y) ( x , y ) → y f ( x / y ) , y>0 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)) ( x , y ) → g ( y ) f ( x / g ( y ) ) , with suitable assumptions on g g . Here, this generalized perspective is shown to be a powerful tool for the analysis of convexity properties of parametrized families of matrix functions