Abstract

Any finite, separately convex, positively homogeneous function on R2\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 perspective of a convex function ff, namely, the function (x,y)→yf(x/y)(x,y)\to yf(x/y), y>0y>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)→g(y)f(x/g(y))(x,y) \to g(y)f(x/g(y)), with suitable assumptions on gg. Here, this generalized perspective is shown to be a powerful tool for the analysis of convexity properties of parametrized families of matrix functions

Author information

Contact details are reproduced from the original publication and may be historical.

Bernard Dacorogna

Section de Mathématiques, Ecole Polytechnique Fédérale, 1015 Lausanne, Switzerland

bernard.dacorogna@epfl.ch

References

  1. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer, Berlin (1989).https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  2. B. Dacorogna: On rank-one convex functions which are homogeneous of degree one, in: Calculus of Variations, Applications and Computations (Pont-à-Mousson, 1994), C. Bandle et al. (ed.), Pitman Research Notes in Mathematics Series 326, Longman Science and Technical, Harlow (1995) 84–99.Google Scholar
  3. B. Dacorogna, J.-P. Haeberly: On convexity properties of homogeneous functions of degree one, Proc. R. Soc. Edinb., Sect. A, Math. 126 (1996) 947–956.https://doi.org/10.1017/S0308210500023180Google Scholar
  4. B. Dacorogna, H. Koshigoe: On the different notions of convexity for rotationally invariant functions, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 2(2) (1993) 163–184.https://doi.org/10.5802/afst.762Google Scholar
  5. B. Dacorogna, P. Marcellini: Implicit Partial Differential Equations, Birkhäuser, Boston (1999).https://doi.org/10.1007/978-1-4612-1562-2Google Scholar
  6. B. Dacorogna, P. Maréchal: A note on spectrally defined polyconvex functions, in: Proceedings of the Workshop “New Developments in the Calculus of Variations (Sannio, 2005)”, Edizioni Scientifiche Italiane (2006), 259–274.Google Scholar
  7. B. Dacorogna, P. Maréchal: Convex SO(N) × SO(n)-invariant functions and refinements of von Neumann’s inequality, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 16(1) (2007) 71–89.https://doi.org/10.5802/afst.1139Google Scholar
  8. J.-B. Hiriart-Urruty, C. Lemaréchal: Convex Analysis and Minimization Algorithms, I and II, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  9. P. Maréchal: On the convexity of the multiplicative potential and penalty functions and related topics, Math. Program. 89A (2001) 505–516.https://doi.org/10.1007/PL00011412Google Scholar
  10. P. Maréchal: On a functional operation generating convex functions. I: Duality, J. Optimization Theory Appl. 126(1) (2005) 175–189.https://doi.org/10.1007/s10957-005-2667-0Google Scholar
  11. P. Maréchal: On a functional operation generating convex functions. II: Algebraic properties, J. Optimization Theory Appl. 126(2) (2005) 357–366.https://doi.org/10.1007/s10957-005-4721-3Google Scholar
  12. P. Maréchal: On a class of convex sets and functions, Set-Valued Anal. 13 (2005) 197–212.https://doi.org/10.1007/s11228-004-4110-3Google Scholar
  13. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar

Suggested citation

B. Dacorogna, P. Maréchal. “The Role of Perspective Functions in Convexity, Polyconvexity, Rank-One Convexity and Separate Convexity.” Journal of Convex Analysis 15 (2008), No. 2, 271–284. https://doi.org/10.68381/jca15019

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.