On Sets Defining Few Ordinary Planes

On Sets Defining Few Ordinary Planes Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than $$Kn^2$$ K n 2 for some $$K=o(n^{{1}/{7}})$$ K = o ( n 1 / 7 ) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than $$\frac{1}{2}n^2-cn$$ 1 2 n 2 - c n then, for n sufficiently large, S is either a regular prism, a regular anti-prism, a regular prism with a point removed or a regular anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than $$Kn^2$$ K n 2 for some $$K=o(n^{{1}/{7}})$$ K = o ( n 1 / 7 ) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete & Computational Geometry Springer Journals

On Sets Defining Few Ordinary Planes

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Mathematics; Combinatorics; Computational Mathematics and Numerical Analysis
ISSN
0179-5376
eISSN
1432-0444
D.O.I.
10.1007/s00454-017-9935-2
Publisher site
See Article on Publisher Site

Abstract

Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than $$Kn^2$$ K n 2 for some $$K=o(n^{{1}/{7}})$$ K = o ( n 1 / 7 ) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than $$\frac{1}{2}n^2-cn$$ 1 2 n 2 - c n then, for n sufficiently large, S is either a regular prism, a regular anti-prism, a regular prism with a point removed or a regular anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than $$Kn^2$$ K n 2 for some $$K=o(n^{{1}/{7}})$$ K = o ( n 1 / 7 ) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four.

Journal

Discrete & Computational GeometrySpringer Journals

Published: Sep 19, 2017

References

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