# 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

, Volume 60 (1) – Sep 19, 2017
34 pages

/lp/springer_journal/on-sets-defining-few-ordinary-planes-duW0PJATn0
Publisher
Springer Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations