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Source. Theorem 1, p. 3, of Konrad J. Swanepoel, Unit distances and diameters in Euclidean spaces, Discrete Comput. Geom. 41 (2009), no. 1, 1--27, doi:10.1007/s00454-008-9082-x; labels and pages are those of arXiv:0707.0213v1 (2 July 2007), the version named on the source card; the proof occupies Sections 5-7, pp. 9-23.

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the printed pages. The proof was read for structure only. Nothing here is independently reviewed.

Statement

Setting. Extremal sets, $u_d(n)$, $M_d(n)$ and Lenz configurations are as in the paper's definitions (pp. 1-3): points on $\lfloor d/2\rfloor$ concentric circles in mutually orthogonal planes, with one circle replaced by a $2$-sphere in a $3$-dimensional summand when $d$ is odd, the radii satisfying $r_i^2+r_j^2=1$ for $i\ne j$.

Theorem 1 (p. 3, quoted). "For each $d\geq 4$ there exists $N(d)$ such that all extremal sets of $n\geq N(d)$ points (with respect to unit distances or diameters) are Lenz configurations."

$N(d)$ is not made explicit. For odd $d$ the conclusion is the strong form of a Lenz configuration (the sphere together with circles, all on the prescribed radii), which the paper reaches in two steps described below.

Proof pointer

Section 7 shows from the stability theorems that extremal sets are, for large $n$, Lenz configurations in a weaker sense: Proposition 19 (p. 20) for even $d\ge4$, Theorem 20 (p. 21) for odd $d\ge7$ and Theorem 21 (p. 22) for $d=5$, each applying Theorem 4 or Theorem 5 and then using extremality, by comparing each exceptional point with a new point placed on one of the circles (for diameters, placed so the diameter does not grow), to put the exceptional points on the configuration. For even $d$ this is already the conclusion. For odd $d$, a weak Lenz configuration lets every factor be a $2$-sphere ($\Sigma_1\cup\cdots\cup\Sigma_p$ for $d\ge7$, p. 11; a variant for $d=5$, pp. 13-14), and Section 5 shows that an optimised weak Lenz configuration is strong for large $n$: Propositions 13 (p. 11) and 14 (p. 12) for $d\ge7$, Propositions 15 (p. 14) and 16 (p. 15) for $d=5$; the unit-distance cases, Propositions 13 and 15, use the $O(m^{4/3})$ bound for unit distances on a $2$-sphere (Lemma 7(d), p. 5).

Dependencies

Theorem 4 and Theorem 5 (stability); Lemma 7 (p. 5) on circles and $2$-spheres, whose part (d) is the bound of Clarkson et al. with the lower bound of Erdős, Hickerson and Pach, cited and not proved here; Lemma 8 (p. 9) on orthogonality of mutually unit-distant sets, whose proof the paper omits as easy.

Bears on

• Problem 223: since the problem's $f_d(n)$ is the paper's $M_d(n)$, the theorem says that for every $d\ge4$ and $n\ge N(d)$ every $n$-point set of diameter one attaining $f_d(n)$ is a Lenz configuration. The exact value follows in Corollary 3.
• Problem 1085: for every $d\ge4$ and $n\ge N(d)$ every $n$-point set attaining the problem's $f_d(n)=u_d(n)$ is a Lenz configuration. This gives the exact value for even $d\ge6$ (Corollary 2); for odd $d\ge5$ the paper says the exact value would follow from the maximum number of unit distances among $m$ points on a $2$-sphere, of radius $1/\sqrt2$ for $d\ge7$ and of arbitrary radius for $d=5$, which it does not determine (pp. 3, 11 and 14). The theorem says nothing about $d=2$ or $d=3$.