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Source. Theorem 3.11, p. 16, of George B. Purdy and Justin W. Smith, Lines, circles, planes and spheres, arXiv:0907.0724 (2009); Discrete Comput. Geom. 44 (2010), no. 4, 860--882, doi:10.1007/s00454-010-9270-3. Labels and pages here are those of arXiv v1 (3 July 2009), whose pagination differs from the journal's; the edition is named on the source card.

Read depth. Claims checked: the statement and its hypotheses were read clause by clause on the page image of the print; the proof was not checked.

Statement

Let $S$ be a set of $n$ points in $\mathbb R^3$, no three collinear and at most $n-k$ coplanar. If

$$n\ \ge\ g(k)=54k^2+\frac92k,$$

then the total number of planes determined by $S$ is at least

$$1+k\binom{n-k}{2}-\binom k2\Bigl(\frac{n-k}{2}\Bigr).$$

The paper calls the theorem a generalization to three dimensions of a theorem of Kelly and Moser (p. 16). By Lemma 3.9 (p. 14), the same bound holds without a condition on $n$ when exactly $n-k$ of the points are coplanar, so the bound is attained when $k=1$.

Proof pointer

Pages 16--17. Calling the number of determined planes through a pair of points its degree, the proof splits into two cases. If more than $n/2$ pairs have degree below $6k$, two of them share a point, and the plane through the three points involved misses fewer than $36k^2$ points of $S$; the bound then follows from Lemma 3.9 and a study of the bound as a cubic in $k$. Otherwise at least $\frac12n(n-2)$ pairs have degree at least $6k$, and Lemma 3.10 (p. 15), a consequence of Melchior's inequality, gives the bound.

Dependencies

Lemmas 3.9 and 3.10 of the paper, and through Lemma 3.10 the paper's Theorem 3.5 (p. 12), which applies Melchior's inequality to projections.

Consequences in the paper

• Corollary 3.14 (p. 20): a set of $n\ge59$ points in $\mathbb R^3$, no three collinear and not all coplanar, determining $m$ planes and $t$ lines, has $m-t+n\ge2$. The case $k=1$ of the theorem gives $m\ge1+\binom{n-1}{2}$ for $n\ge59$, which Erdős and Purdy had proved for $n\ge552$ (p. 20).
• Corollary 3.15 (p. 21): a set of $n\ge225$ points in $\mathbb R^3$, no three collinear and no $n-1$ coplanar, determining $m$ planes and $t$ lines, has $m\ge t$.

The paper relates both corollaries to conjectures of Purdy (pp. 3, 21), and notes that Erdős asked for sufficient conditions for $m\ge t$ (p. 21).

Bears on

None of the problem pages directly.