Extremal points of a functional

on the set of convex functions

Authors:
T. Lachand-Robert and M. A. Peletier

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1723-1727

MSC (1991):
Primary 49K99

DOI:
https://doi.org/10.1090/S0002-9939-99-05209-0

Published electronically:
February 11, 1999

MathSciNet review:
1646197

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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

**1.**F. Brock, V. Ferone and B. Kawohl,*A symmetry problem in the calculus of variations*, Calculus of Variations and Partial Differential Equations,**4**(1996), pp. 1723-1727.MR**97i:49002****2.**G. Buttazzo, V. Ferone and B. Kawohl,*Minimum problems over sets of concave functions and related questions*, Math. Nachrichten,**173**(1993), pp. 71-89.MR**96b:49005****3.**I. Ekeland and R. Temam,*Convex Analysis and Variational Problems*, North-Holland (1972). MR**57:3931b****4.**T. Lachand-Robert and M. A. Peletier,*An example of non-convex minimization and an application to Newton's problem of the body of least resistance*, in preparation.**5.**P. Marcellini,*Nonconvex Integrals of the Calculus of Variations*, Proceedings of `Methods of Nonconvex Analysis', Varenna 1989, ed. A. Cellina, Lecture Notes in Math.,**1446**, Springer (1990), pp. 1723-1727. MR**91j:49002****6.**J.-C. Rochet and P. Choné,*Ironing, Sweeping and Multidimensional screening*, to appear in Economica.

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Additional Information

**T. Lachand-Robert**

Affiliation:
Université Pierre et Marie Curie, Laboratoire d’Analyse Numérique, 75252 Paris Cedex 05, France

Email:
lachand@ann.jussieu.fr

**M. A. Peletier**

Affiliation:
University of Bath, Claverton Down, Bath BA2 7AY United Kingdom

Email:
M.A.Peletier@bath.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-99-05209-0

Keywords:
Extremal points,
convexity constraint,
non-convex minimization

Received by editor(s):
September 10, 1997

Published electronically:
February 11, 1999

Additional Notes:
Part of this work was carried out during a visit of the second author to Université Pierre et Marie Curie under the contract of the European Union 921 CHRX CT 94.

Communicated by:
Jeffrey B. Rauch

Article copyright:
© Copyright 1999
American Mathematical Society