A SPHERICAL MAPPING AND BORSUK CONJECTURE IN RIEMANNIAN AND NON-EUCLIDEAN SPACES

Article Sidebar

PDF
Published: Jun 1, 1994
DOI: https://doi.org/10.5556/j.tkjm.25.1994.4436
Keywords:
Convex bodies in manifolds spherical mapping Borsuk conjecture

Main Article Content

BORIS V. DEKSTER
Department of Mathematics, Mount Allison University, Sackville, N.B. EOA 3CO, Canada.

Abstract




We introduce an analog of the spherical mapping for convex bodies in a Riemannian $n$-manifold, and then use this construction to prove the Borsuk conjecture for some types of such bodies. The Borsuk conjecture is that each bounded set $X$ in the Euclidean $n$-space can be covered by $n +1$ sets of smaller diameter. The conjecture was disproved recently by Kahn and Kalai. However Hadwiger proved the Borsuk conjecture under the additional assumption that the set $X$ is a smooth convex body. Here we extend this result to convex bodies in Riemannian manifolds under some further restrictions.




Article Details

How to Cite
DEKSTER, B. V. (1994). A SPHERICAL MAPPING AND BORSUK CONJECTURE IN RIEMANNIAN AND NON-EUCLIDEAN SPACES. Tamkang Journal of Mathematics, 25(2), 149–155. https://doi.org/10.5556/j.tkjm.25.1994.4436
Issue
Vol. 25 No. 2 (1994)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

V. Boltjansky and I. Gohberg, Results and problems in Combinatorial Geometry, Cambridge Uni- versity Press, Cambridge-London-New York, 1985.

Yu. D. Burago and V. A. Zalgaller, "Convex sets in Riemannianspaces of non-negative curvature." Russian Math. Surveys, 32 (1977) no. 3, 1-57.

B. V. Dekster, "Bodies of constant width in Riemannian manifolds and spaces of constant curvature," In: Applied Geometry and Discrete Mathematics, Victor Klee Festschrift, DIMACS series, V. 4, (1991), edited by Peter Gritzman and Bernd Strumfels, copublished by AMS and ACM, 181-192.

B. V. Dekster," Width and breadth for convex bodies in Riemanian manifolds." Archiv der Math- ematik, Vol. 58, (1992), 190-198.

B. V. Dekster, "Double normals characterize bodies of constant width in Riemannian manifolds." In: Geometric analysis and nonlinear partial differential equations. Editor: I. Ya. Bakelman. Marcel Dekker , New York-Basel-Hong Kong, (1993), 187-201.

H. Hadwiger, "Uberdeckung eine Menge durch Mengen kleineren Durchmessers." Comment. Math. Helv., 18, (1945/ 46), 73-75.

H.Hadwiger,Mitteilung betreffend meiner Note:"Uberdeckung eine Menge durch Mengen kleineren Duchmessers," Comment. Math. Helv., 19, (1946/47), 72-73.

H. Hadwiger, "Uber die Zerstiickung eines Eikorpers," Math. Z. 51, (1947), 161-165.

J. Kahn, and G. Kalai, "A counterexample to Borsuk's conjecture," Bulletin of AM S, vol.29, no.1 (1993), 60-62.