On sets defining few ordinary planes
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hdl:2117/111528
Tipus de documentArticle
Data publicació2017-09-19
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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 (Formula presented.) for some (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) for some (Formula presented.) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four.
CitacióBall, S. On sets defining few ordinary planes. "Discrete and computational geometry", 19 Setembre 2017, vol. 60, nº 1, p. 1-34.
ISSN0179-5376
Versió de l'editorhttps://link.springer.com/article/10.1007%2Fs00454-017-9935-2
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