Abstract
A set is said to be H-convex if it can be represented by an intersection of a family of closed half-spaces whose outer normals belong to a given subset of the set H of the unit sphereS n−1⊂R. On the basis of Helly’s theorem for H-convex sets recently obtained by us, we prove in this note certain extensions of Blaschke’s theorem (on the radius of an inscribed sphere) and of several other well-known theorems of combinatorial geometry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
V. G. Boltyanskii, “Helly’s theorem for H-convex sets,” Dokl. Akad. Nauk SSSR,226, No. 2, 249–252 (1976).
I. M. Yaglom and V. G. Boltyanskii, Convex Figures [in Russian], Gostekhizdat, Moscow-Leningrad (1951).
L. Danzer, B. Grünbaum, and V. Klee, Helly’s Theorem, Proc. Symposium Pure Mathematics, Amer. Math. Society, Providence, R. I. (1963).
V. G. Boltyanskii, “On certain classes of convex sets,” Dokl. Akad. Nauk SSSR,226, No. 1, 19–22 (1976).
V. G. Boltyanskii and P. S. Soltan, “On starlike sets,” Izv. Akad. Nauk MoldSSR,10, No. 3, 7–11 (1976).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 117–124, January, 1977.
Rights and permissions
About this article
Cite this article
Boltyanskii, V.G. Several theorems of combinatorial geometry. Mathematical Notes of the Academy of Sciences of the USSR 21, 64–68 (1977). https://doi.org/10.1007/BF02317039
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02317039