Abstract
We prove that a finite family ℬ={B 1,B 2, ...,B n } of connected compact sets in ℝd has a hyperplane transversal if and only if for somek there exists a set of pointsP={p 1,p 2, ...,p n } (i.e., ak-dimensional labeling of the family) which spans ℝk and everyk+2 sets of ℬ are met by ak-flat consistent with the order type ofP. This is a common generalization of theorems of Hadwiger, Katchalski, Goodman-Pollack and Wenger.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Danzer, B. Grünbaum, andV. Klee: ‘Helly's theorem and its relatives’,Convexity, Proc. Symp. Pure Math.,7 (1963), 100–181. Amer. Math. Soc.
J.Eckhoff: Radon's theorem,Contributions to Geometry, ed. J. Tölke and J. M. Wills, Birkhäuser (1979), 164–185.
J. E. Goodman, andR. Pollack: Multidimensional sorting,SIAM J. Computing,12 (1983), 484–507.
J. E.Goodman, and R.Pollack: Hadwiger's transversal theorem in higher dimensions,Journal of Am. Math Soc.,1 (2).
B. Grünbaum: On a theorem of L. A. Santaló,Pacific J. Math.,5 (1955), 351–359.
H. Hadwiger: Über Eibereiche mit gemeinsamer Treffgeraden,Portugal Math.,6 (1957), 23–29.
H.Hadwiger, H.Debrunner, and V.Klee:Combinatorial Geometry in the Plane, Holt, Rinehart and Winston (1964).
W. R. Hare, andJ. W. Kenelly: Characterizations of Radon partitions,Pacific J. Math.,36 (1971), 159–146.
E. Helly: Über mengen konvexer Körper mit gemeinschaftlichen Punkten,Jber. Deutsche Math. Verein,32 (1923), 175–176.
M. Katchalski: Thin sets and common transversals,J. of Geometry,14 (1980), 103–107.
P. Kirchberger: Über Tschebyscfefsche Annäherungsmethoden,Math. Ann.,57 (1903), 509–540.
J. R.Munkres:Elements of Algebraic Topology, Addition Wesley (1984).
L. A. Santaló: A theorem on sets of parallelepipeds with parallel edges,Publ. Inst. Mat. Univ. Nac. Lioral,2 (1940), 49–60.
P. Vincensini: Figures convexes et varieté de l'escape àn dimensions,Bull. Sciences Math., (2)59 (1935), 163–174.
R.Wenger: A generalization of Hadwiger's Theorem to intersecting sets,Discrete and Comp. Geom., to appear.
Author information
Authors and Affiliations
Additional information
Supported in part by NSF grant DMS-8501947 and CCR-8901484, NSA grant MDA904-89-H-2030, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648.
Supported by the National Science and Engineering Research Council of Canada and DIMACS.