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Source. Sections 1.1 (pp. 1-2), 1.2 (p. 2) and 2 (pp. 2-4), with the definitions on pp. 1-3, 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.

Read depth. Claims checked: each definition was read clause by clause on the printed pages. Nothing here is independently reviewed.

Statement

• Unit distances (p. 1). For a finite $S\subset\mathbb R^d$, $u(S)$ is the number of pairs of points of $S$ at distance $1$, and $u_d(n)=\max\{u(S):S\subset\mathbb R^d,\ |S|=n\}$.
• Diameters (p. 2). A pair of points of a finite $S\subset\mathbb R^d$ is a diameter when their distance equals the diameter of $S$; $M(S)$ is the number of diameters of $S$, and $M_d(n)=\max\{M(S):S\subset\mathbb R^d,\ |S|=n\}$.
• Lenz configuration, even $d\ge4$ (p. 2). Put $p=d/2$ and take any orthogonal decomposition $\mathbb R^d=V_1\oplus\cdots\oplus V_p$ into $2$-dimensional subspaces. In each $V_i$ let $C_i$ be the circle with centre the origin $o$ and radius $r_i$, where $r_i^2+r_j^2=1$ for all distinct $i,j$. A Lenz configuration is any translate of a finite subset of $\bigcup_{i=1}^pC_i$. For $d\ge6$ the radius condition forces every $r_i=1/\sqrt2$; for $d=4$ only $r_1^2+r_2^2=1$ is required.
• Lenz configuration, odd $d\ge5$ (p. 3). Put $p=\lfloor d/2\rfloor$ and take any orthogonal decomposition $\mathbb R^d=V_1\oplus\cdots\oplus V_p$ with $V_1$ of dimension $3$ and $V_2,\ldots,V_p$ of dimension $2$. Let $\Sigma$ be the sphere in $V_1$ with centre $o$ and radius $r_1$, and for $i=2,\ldots,p$ let $C_i$ be the circle in $V_i$ with centre $o$ and radius $r_i$, where $r_i^2+r_j^2=1$ for all distinct $i,j$. A Lenz configuration is any translate of a finite subset of $\Sigma\cup\bigcup_{i=2}^pC_i$. For $d\ge7$ every $r_i=1/\sqrt2$. The paper notes that this is what its later sections call a strong Lenz configuration, as against the weak Lenz configurations used inside the proofs (Sections 5.3 and 5.4, pp. 11 and 13-14).
• Extremal set (p. 3). A set $S$ of $n$ points of $\mathbb R^d$ is extremal with respect to unit distances when $u(S)=u_d(n)$, and extremal with respect to diameters when $M(S)=M_d(n)$.

In a Lenz configuration any two points on different circles (or on the sphere and a circle) are at distance $1$; with $p=\lfloor d/2\rfloor$ circles of radius $1/\sqrt2$ and $n/p+O(1)$ points on each, this is Lenz's construction of $\frac{p-1}{2p}n^2-O(1)$ unit distances recalled on p. 1.

Proof pointer

Definitions; nothing to prove. Section 4 (p. 5) adds the unit distance graph, the counts $u(x,S)$ and $u(A,B)$, and the convention that when diameters are counted the diameter is scaled to $1$, so that $M(S)=u(S)$.

Dependencies

None.

Bears on

• Problem 223: the problem's $f_d(n)$, the most pairs at distance one among $n$ points of diameter one in $\mathbb R^d$, is the paper's $M_d(n)$, since scaling a set to diameter $1$ turns its diameters into its pairs at distance $1$.
• Problem 1085: the problem's $f_d(n)$ is the paper's $u_d(n)$.