Abstract
We investigate those spherical point sets which, relative to the Hausdorff metric, give local minima of the diameter function, and obtain estimates which, in principle, justify computer-generated configurations. We test the new sets against two classical conjectures: Borsuk's and McMullen's.
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L. J. Billera and C. W. Lee, A proof of the sufficiency of McMullen's conditions forf-vectors of simplicial convex polytopes,J. Combin. Theory Ser. A 31 (1981), 237–255.
K. Borsuk, Drei Sätze über dien-dimensionale euklidische Sphäre,Fund. Math. 20 (1933), 177–190.
J. Cheeger and D. G. Ebin,Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975.
L. Fejes Tóth,Lagerungen in der Ebene auf der Kugel und im Raum, Zweite Auflage, Springer-Verlag, Berlin, 1972.
M. Gromov, Curvature, diameter and Betti numbers,Comment. Math. Helv. 56 (1981), 179–195.
M. Gromov, Filling Riemannian manifolds,J. Differential Geom.,18 (1983), 1–147.
K. Grove and K. Shiohama, A generalized sphere theorem,Ann. of Math. 106 (1977), 201–211.
B. Grünbaum, A proof of Vázsonyi's conjecture,Bull. Res. Council Israel A 6 (1956), 77–78.
A. Heppes, Beweis einer Vermutung von A. Vázsonyi,Acta Math. Acad. Sci. Hungar. 7 (1956), 463–466.
M. Katz, The filling radius of two-point homogeneous spaces,J. Differential Geom. 18 (1983), 505–511.
L. Lovász, Self-dual polytopes and the chromatic number of distance graphs on the sphere,Acta Sci. Math. 45 (1983), 317–323.
P. McMullen, The numbers of faces of simplicial polytopes,Israel J. Math. 9 (1971), 559–570.
P. McMullen and G. C. Shephard,Convex Polytopes and the Upper Bound Conjecture, London Mathematical Society Lecture Note Series, Vol. 3, Cambridge University Press, London, 1971.
J. Molnár, Über eine Übertragung des Hellyschen Satzes in sphärische Räume,Acta Math. Acad. Sci. Hungar. 8 (1957), 315–318.
R. Stanley, The number of faces of a simplicial convex polytope,Adv. in Math. 35 (1980), 236–238.
S. Straszewicz, Sur un problème géométrique de P. Erdös,Bull. Acad. Polon. Sci. Cl. III 5 (1957), 39–40.
D. W. Walkup, The lower bound conjecture for 3- and 4-manifolds,Acta Math. 125 (1970), 75–107.
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Supported in part by NSF Grant DMS 8602645.
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Katz, M. Diameter-extremal subsets of spheres. Discrete Comput Geom 4, 117–137 (1989). https://doi.org/10.1007/BF02187719
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DOI: https://doi.org/10.1007/BF02187719