Optimal Configurations of Finite Sets in Riemannian 2-Manifolds

Geometriae Dedicata

Volumn 84, Issue 1-3, 2001, Pages 271-320

  Gruber, Peter M ^a  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

Almost regular hexagonal configurations; Best approximating polytopes; Minimum isoperimetric quotient; Optimal integration formulae; Stability; Sums of moments

Indexed keywords

EID: 0010091682     PISSN: 00465755     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1010358407868     Document Type: Article

References (32)

1. Babenko, V. F.: On the optimal error bound for cubature formulas on certain classes of continuous functions, Anal. Math. 3 (1977), 3-9.
2. Bollobas, B. and Stern, N.: The optimal structure of market areas, J. Econom. Theory 4 (1972), 174-179.
3. Böröczky, K. and Ludwig, M.: Approximation of convex bodies and a momentum lemma for power diagrams, Monatsh. Math. 127 (1999), 101-110.
4. Brauner, H.: Differentialgeometrie, Vieweg, Braunschweig, 1981.
5. Diskant, V. I.: Making precise the isoperimetric inequality and stability theorems in the theory of convex bodies, Trudy Inst. Mat. 14 (1989), 98-132.
6. Diskant, V. I.: Generalizations of a theorem of Lindelöf, Mat. Fiz. Anal. Geom. 4 (1997), 59-64.
7. Fejes Tóth, G.: Sum of moments of convex polygons, Acta Math. Acad. Sci. Hungar. 24 (1973), 417-421.
8. Fejes Tóth, L.: The isepiphan problem for n-hedra, Amer. J. Math. 70 (1948), 174-180.
9. Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, 1953, 2nd ed. 1972.
10. Fejes Tóth, L.: Sur la représentation d'une population infinie par un nombre fini d'éléments, Acta Math. Acad. Sci. Hungar. 10 (1959), 299-304.
11. Fejes Tóth, L.: On the isoperimetric property of the regular hyperbolic tetrahedra, Publ. Math. Inst. Hung. Acad. Sci. 8 (1963), 53-57.
12. Florian, A.: Integrale auf konvexen Mosaiken, Period. Math. Hungar. 6 (1975), 23-38.
13. Glasauer, S., Gruber, P. M.: Asymptotic estimates for best and stepwise approximation of convex bodies III, Forum Math. 9 (1997), 383-404.
14. Gruber, P. M.: Volume approximation of convex bodies by circumscribed polytopes, Discrete Math. Theor. Comput. Sci. 4 (1991), 309-317.
15. Gruber, P. M.: Asymptotic estimates for best and stepwise approximation of convex bodies II Forum. Math. 5 (1993), 521-537.
16. Gruber, P. M.: Aspects of approximation of convex bodies, In: Handbook of Convex Geometry A, North-Holland, Amsterdam, 1993, pp. 319-345.
17. Gruber, P. M.: Comparisons of best and random approximation of convex bodies by polytopes, Rend. Sem. Mat. Univ. Palermo, Ser. II, Suppl. 50 (1997), 189-216.
18. Gruber, P. M.: Asymptotic estimates for best and stepwise approximation of convex bodies IV, Forum Math. 10 (1998), 665-686.
19. Gruber, P. M.: A short analytic proof of Fejes Tóth's theorem on sums of moments, Aequat. Math. 58 (1999), 291-295.
20. Gruber, P. M.: Optimal arrangement of finite point sets in Riemannian 2-manifolds, Trudy Mat. Inst. Steklov 225 (1999), 160-167, Proc. Steklov Inst. Math. 225 (1999) 148-155.
21. Gruber, P. M.: Optimal arrangement of finite point sets in Riemannian 2-manifolds, Trudy Mat. Inst. Steklov 225 (1999), 160-167, Proc. Steklov Inst. Math. 225 (1999) 148-155.
22. Haimovich, M. and Magnanti, T. L.: Extremum properties of hexagonal partitioning and the uniform distribution in Euclidean location, SIAM. J. Discrete Math. 1 (1988), 50-64.
23. Hajós, G.: Über den Durchschnitt eines Kreises und eines Polygons, Ann. Univ. Sci. Budapest Eötvös, Sect. Math. 11 (1968), 137-144.
24. Hlawka, E.: The Theory of Uniform Distribution, A B Academic Publishers, Berkhamsted, 1984.
25. Imre, M.: Kreislagerungen auf Flächen konstanter Krümmung, Acta Math. Acad. Sci. Hungar. 15 (1964), 115-121.
26. Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Wiley, New York 1969.
27. Leichtweiss, K.: Affine Geometry of Convex Bodies, Johann Ambrosius Barth-Verlag, Heidelberg, 1998.
28. Papadimitriou, C. H.: Worst-case and probabilistic analysis of a geometric location problem, SIAM. J. Comput. 10 (1981), 542-557.
29. Schneider, R.: Affine-invariant approximation by convex polytopes, Studia Sci. Math. Hungar. 21 (1986), 401-408.
30. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.
31. Shiohama, K. and Tanaka, M.: Cut loci and distant spheres on Alexandrov surfaces, In: Besse, A. L., (ed.), Actes de la table ronde de geometrie differentielle en l'honneur de Marcel Berger, Luminy, France, 1992, Soc. Math. France. Semin. Congr. 1 (1996), pp. 531-559.
32. Thompson, A. C.: Minkowski Geometry, Cambridge University Press, Cambridge 1996.