Separation of a finite set in ℝd by spanned hyperplanes

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

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    B. Grünbaum: Partitions of mass-distributions and of convex bodies by hyperplanes,Pacific J. Math. (1960), 1257–1261.

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    Y. S. Kupitz: On the existence ofk-points, in preparation.

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    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

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Kupitz, Y.S. Separation of a finite set in ℝd by spanned hyperplanes. Combinatorica 13, 249–258 (1993). https://doi.org/10.1007/BF01202351

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AMS subject classification code (1991)

  • 52 A 35
  • 52 C 10