Abstract
A question of the following kind will concern us here: what is the minimal numbern, ensuring that any spanning set ofn points in 3-space spans a plane, every open side of which contains at least, say, 1000 points of the set. The answer isn=4001 (see Theorem 2.1 below).
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References
B. Grünbaum: Partitions of mass-distributions and of convex bodies by hyperplanes,Pacific J. Math. (1960), 1257–1261.
Y. S. Kupitz: On the existence ofk-points, in preparation.
Y. S. Kupitz: On a generalization of Sylvester's theorem,Discrete & Comput. Geometry 7 (1992), 87–103.
Y. S. Kupitz:k-supporting spanned hyperplanes of a finite set in ℝd,J. Comb. Theory, Series A, forthcoming.
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This is a part of a Ph.D. thesis, with the same title, supervised by Professor Micha A. Perles in the Hebrew University of Jerusalem