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Source. Theorem 4.2, p. 24, 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 $n\ge5$ points in $\mathbb R^3$, not all cospherical or coplanar, no four cocircular and no three collinear. Then there are at least $\epsilon\binom n3$ spheres incident to exactly four points of $S$, where

$$\epsilon=\frac9{208}.$$

Proof pointer

Pages 24--25. The proof inverts $S\setminus\{p\}$ in a sphere about a point $p\in S$; by Lemma 4.1 (p. 23) the inverted set keeps the hypotheses, and three-point planes of the image that avoid $p$ correspond to four-point spheres through $p$. Theorem 3.8 supplies the three-point planes, and two cases on how many of them pass through $p$, the second settled by a count of pairs, give at least $\frac3{104}(n-1)(n-2)$ four-point spheres through $p$. Summing over $p$ counts each sphere four times. The case of $n-1$ coplanar points is handled directly.

Dependencies

Theorem 3.8 and Lemma 4.1 of the paper.

Bears on

None of the problem pages directly.