Discrete & Computational Geometry

, Volume 15, Issue 2, pp 147–167

Equipartition of mass distributions by hyperplanes

• E. A. Ramos
Article

Abstract

We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR d , there arek hyperplanes so that each orthant contains a fraction 1/2 k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2 k −1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2k−1. We are able to prove that Δ(j, k)=⌈j(2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2 n withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5).

Keywords

Mass Distribution Cell Complex Component Function Gray Code Moment Curve

References

1. 1.
N. Alon and D. B. West. The Borsuk-Ulam theorem and bisection of necklaces,Proc. Amer. Math. Soc. 98 (1986), 623–628.
2. 2.
D. Avis, Non-partitionable point sets,Inform. Process. Lett. 19 (1984), 125–129.
3. 3.
D. I. A. Cohen, On the combinatorial antipodal-point lemmas,J. Combin. Theory Ser. B,27 (1979), 87–91.
4. 4.
R. Downey and M. Fellows, Fixed parameter intractability (extended abstract), inProceedings of the 7th Annual IEEE Conference on Structures in Complexity Theory, 1992, pp. 36–49.Google Scholar
5. 5.
E. Fadell and S. Husseini, Anideal-valued cohomological index-theory with applications to Borsuk-Ulam and Bourgin-Yang theorems,Ergodic Theory Dynamical Systems 8 * (1988), 73–85.
6. 6.
K. Fan, A generalization of Tucker's combinatorial lemma with topological applications.Ann. of Math. 56 (1952), 431–437.
7. 7.
C. H. Goldberg and D. B. West, Bisection of circle colorings,SIAM J. Algebraic Discrete Methods 6 (1985) 93–106.
8. 8.
B. Grünbaum, Partitions of mass-distributions and on convex bodies by hyperplanes,Pacific J. Math. 10 (1960), 1257–1261.
9. 9.
H. Hadwiger, Simultane Vierteilung zweier Körper,Arch. Math. (Basel) 17 (1966), 274–278.
10. 10.
M. W. Hirsch, A proof of the nonretractibility of a cell onto its boundary,Proc. Amer. Math. Soc. 14 (1963), 364–365.
11. 11.
C.-Y. Lo, J. Matoušek, and W. Steiger, Algorithms for ham-sandwich cuts.Discrete Comput. Geom. 11 (1994), 433–452.
12. 12.
J. Matoušek, Efficient partition trees,Discrete Comput. Geom. 8 (1992), 315–334.
13. 13.
J. Matoušek, Range searching with efficient hierarchical cuttings,Discrete Comput. Geom. 10 (1993), 157–182.
14. 14.
J. Matoušek, Geometric Range Searching, Report B 93-09, Institute for Computer Science, Department of Mathematics, Freie Universität, Berlin, July 1993.Google Scholar
15. 15.
C. Papadimitriou, On the complexity of the parity argument and other inefficient proofs of existence,J. Computer System Sci. 48 (1994), 498–532.
16. 16.
J. P. Robinson and M. Cohn, Counting sequences,IEEE Trans. Comput. 30 (1981), 17–23.
17. 17.
A. W. Tucker, Some topological properties of disk and sphere, inProceedings of the First Canadian Mathematical Congress, Montreal, 1945, pp. 285–309.Google Scholar
18. 18.
D. G. Wagner and J. West, Construction of uniform Gray codes,Congr. Numer. 80 (1991), 217–223.
19. 19.
D. E. Willard, Polygon retrieval,SIAM J. Comput. 11 (1982), 149–165.
20. 20.
A. C. Yao and F. F. Yao, A general approach tod-dimensional geometric queries, inProceedings of the 17th ACM Annual Symposium on the Theory of Computing, 1985, pp. 163–169.Google Scholar
21. 21.
F. Yao, D. Dobkin, H. Edelsbrunner, and M. Paterson, Partitioning space for range queries,SIAM J. Comput. 18 (1989), 371–384.
22. 22.
C. Zhong, The Borsuk-Ulam theorem on product space, inFixed Point Theory and Applications (Proc. Conf., Halifax, Nova Scotia, June 1991), pp. 362–372, ed. K. K. Tan, World Scientific, Singapore, 1992.Google Scholar