The Role of Perspective Functions in Convexity, Polyconvexity, Rank-One Convexity and Separate Convexity
Journal of convex analysis, Tome 15 (2008) no. 2, pp. 271-284
Voir la notice de l'article provenant de la source Heldermann Verlag
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.
@article{JCA_2008_15_2_JCA_2008_15_2_a5,
author = {B. Dacorogna and P. Mar\'echal},
title = {The {Role} of {Perspective} {Functions} in {Convexity,} {Polyconvexity,} {Rank-One} {Convexity} and {Separate} {Convexity}},
journal = {Journal of convex analysis},
pages = {271--284},
publisher = {mathdoc},
volume = {15},
number = {2},
year = {2008},
url = {http://geodesic.mathdoc.fr/item/JCA_2008_15_2_JCA_2008_15_2_a5/}
}
TY - JOUR AU - B. Dacorogna AU - P. Maréchal TI - The Role of Perspective Functions in Convexity, Polyconvexity, Rank-One Convexity and Separate Convexity JO - Journal of convex analysis PY - 2008 SP - 271 EP - 284 VL - 15 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/JCA_2008_15_2_JCA_2008_15_2_a5/ ID - JCA_2008_15_2_JCA_2008_15_2_a5 ER -
%0 Journal Article %A B. Dacorogna %A P. Maréchal %T The Role of Perspective Functions in Convexity, Polyconvexity, Rank-One Convexity and Separate Convexity %J Journal of convex analysis %D 2008 %P 271-284 %V 15 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/JCA_2008_15_2_JCA_2008_15_2_a5/ %F JCA_2008_15_2_JCA_2008_15_2_a5
B. Dacorogna; P. Maréchal. The Role of Perspective Functions in Convexity, Polyconvexity, Rank-One Convexity and Separate Convexity. Journal of convex analysis, Tome 15 (2008) no. 2, pp. 271-284. http://geodesic.mathdoc.fr/item/JCA_2008_15_2_JCA_2008_15_2_a5/