Convexity and Geodesic Metric Spaces

Authors
Sajad Ranjbar 1
Hadi Khatibzadeh 2
Parviz Ahmadi 2
1 Higher Education Center of Eghlid
2 University of Zanjan
10.52547/mmr.7.1.67
Abstract
In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metric spaces.

./files/site1/files/71/7.pdf
Keywords
 Geodesic
Metric segment
d-convex set
Metric convex hull
d-convex function
Extreme point
Continuity
1. Bielawski R., "Simplicial convexity and its applications", J. Math. Anal. Appl., 127 (1987) 155-161.## 2. Boltyanski V., Martini H., Soltan P. S., "Excursions into Combinatorial Geometry", Universitext, Springer-Verlag, Berlin (1997). ## 3. Cioranescu I., "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems", Mathematics and its Applications, 62. Kluwer Academic Publishers Group, Dordrecht (1990). ## 4. Kutateladze S. S., Rubinov A. M., "Minkowski duality and its applications", Usp. Mat. Nauk, 27 (1972) 137-192. ## 5. Krynski S., "Metrically convex functions in normed spaces", Studia Math. 105 (1993) 1-11. 6. Lucchetti R., "Convexity and Well-posed Problems", CMS Books in Mathematics, Springer, New York (2006). ## 7. Menger K., "Untersuchen ¨uber allgemeine Metrik, I, II, III," Math. Ann., 100 (1928) 75-163. ## 8. Michael E., "Convex structures and continuous selections", Canad. J. Math., 11 (1959) 556-575. ## 9. Pallaschke D., Rolewicz S., "Foundations of Mathematical Optimization", Math. Appl., Vol. 388, Kluwer Academic Publ., Dordrecht-Boston-London (1997). ## 10. Papadopoulos A., "Metric Spaces, Convexity and Nonpositive Curvature", IRMA Lectures in Mathematics and Theoretical Physics, 6. European Mathematical Society (EMS), Zürich, (2005). ## 11. Roberts A.W., Varberg D.E., "Convex Functions, Pure and Applied Mathematics", Vol. 57. Academic Press, New York-London (1973). ## 12. Rudin W., "Functional Analysis", Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York (1991). ## 13. Rubinov A. M., "Abstract Convexity and Global Optimization", Nonconvex Optimization and Its Applications, Vol. 44, Kluwer Academic Publ.(2000). ## 14. Rubinov A. M., "Abstract convexity: Examples and Applications, Optimization", 47 (2000) 1-33. ## 15. Singer I., "Abstract Convex Analysis", Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley Sons, Inc., New York (1997). ## 16. Soltan V. P., "Introduction to Axiomatic Theory of Convexity[in Russian]", Stiinca, Kishiniev (1984). ## 17. Soltan V. P., Soltan P. S., "d-convex functions", Dokl. Akad. Nauk SSSR, Ser. Mat., 249 (1979) 555-568. ## 18. Stone M. H., "Postulates for the barycentric calculus", Ann. Mat. Pura Appl., 29 (1949) 25-30. ## 19. Van De Vel M., "Finite dimensional convex structures I: general results", Topology Appl., 14 (1982) 201-225. ## 20. Van De Vel M., "A selection theorem for topological convex structures", Trans. Amer. Math. Soc., 336 (1993) 463-496. ##