Abstract
Some properties of the spaces of paths are studied in order to define and characterize the local convexity of sets belonging to smooth manifolds and the local convexity of functions defined on local convex sets of smooth manifolds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. De Finetti (1949) ArticleTitleSulle Stratificazioni Convesse Annali di Matematica Pura ed Applicata 30 173–183
W. Fenchel (1951) Convex Cones, Sets, and Functions, Mimeographed Lecture Notes Princeton University Press Princeton, New Jersey
W. Fenchel (1956) ArticleTitleÜber Konvexe Funktionen mit vorgeschriebenen Niveaumannigfaltigkeiten Mathematische Zeitschrift 63 496–506 Occurrence Handle10.1007/BF01187955
A.W. Roberts D.E. Varberg (1973) Convex Functions Academic Press New York, NY
G. Debreu (1954) Representation of a Preference Ordering by a Numerical Function R. Thrall X.X. Coombs X.X. Davis (Eds) Decision Processes. John Wiley and Sons New york, NY
J.P. Crouzeix (1977) Contributions à l’ Étude des Fonctions Quasiconvexes, Thèse Université de Clermont-Ferrand Clermont-Ferrand, France
Y. Kannai (1977) ArticleTitleConcavifiability and Constructions of Concave Utility Functions Journal of Mathematical Economics 4 1–56
Y. Kannai (1981) Concave Utility Functions: Existence, Constructions, and Cardinality S. Schaible W.T. Ziemba (Eds) Generalized Concavity in Optimization and Economics. Academic Press New York, NY 543–611
T. Rapcsák (1991) ArticleTitleOn Pseudolinear Functions European Journal of Operational Research 50 353–360
T. Rapcsák (1997) ArticleTitleAn Unsolved Problem of Fenchel Journal of Global Optimization 11 207–217
T. Rapcsák (1997) Smooth Nonlinear Optimization in Rn Kluwer Academic Publishers Dordrecht, Holland
M. Avriel W.E. Diewert S. Schaible I. Zang (1988) Generalized Concavity Plenum Press New York, NY
F. Giannessi (1984) ArticleTitleTheorems of the Alternative and Optimality Conditions Journal of Optimization Theory and Applications 42 331–365 Occurrence Handle10.1007/BF00935321
Giannessi F., Rapcsák T. (1995). Images, Separation of Sets, and Extremum Problems, Recent Trends in Optimization Theory and Applications. World Scientific, Singapore Republic of Singapore, Vol. 5, pp. 79-106.
G. Mastroeni T. Rapcsák (2000) ArticleTitleOn Convex Generalized Systems Journal of Optimization Theory and Applications 3 605–627 Occurrence Handle10.1023/A:1004641726264
C. Udriste (1994) Convex Functions and Optimization Methods on Riemannian Manifolds Kluwer Academic Publishers Dordrecht, Holland
J.H.C. Whitehead (1932) ArticleTitleConvex Regions in the Geometry of Paths Quarterly Journal of Mathematics 3 33–42
R. Pini (1994) ArticleTitleConvexity along Curves and Invexity Optimization 29 301–309
S. Komlósi (1983) Second-Order Characterization of Pseudoconvex and Strictly Pseudoconvex Functions in Terms of Quasi-Hessians F. Forgó (Eds) Contributions to the Theory of Optimization. Department of Mathematics, Karl Marx University of Economics Budapest, Hungary 19–46
T. Rapcsák (2005) ArticleTitleFenchel Problem of Level Sets Journal of Optimization Theory and Applications 127 177–191
Author information
Authors and Affiliations
Additional information
This paper is dedicated to the memory of Guido Stampacchia. This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by CNR, Rome, Italy.
Rights and permissions
About this article
Cite this article
Rapcsák, T. Local Convexity on Smooth Manifolds. J Optim Theory Appl 127, 165–176 (2005). https://doi.org/10.1007/s10957-005-6398-z
Issue Date:
DOI: https://doi.org/10.1007/s10957-005-6398-z