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Source. D. Burt, E. Goldstein, S. Manski, S. J. Miller, E. A. Palsson and H. Suh, Crescent configurations, arXiv:1509.07220v1 [math.CO] (24 September 2015); Definitions 1.1 and 1.2 on p. 2. The copy read is identified on the source card.

Read depth. Claims checked: both definitions were read clause by clause on the page images. Nothing here is independently reviewed.

Statement

Definition 1.1 (General Position, p. 2). Points in $\mathbb R^d$ are in general position when no $d+1$ of them lie on one hyperplane and no $d+2$ of them lie on one hypersphere.

Definition 1.2 (Crescent Configuration, p. 2). "We say $n$ points are in crescent configuration (in $\mathbb R^d$) if they lie in general position in $\mathbb R^d$ and determine $n-1$ distinct distances, such that for every $1\le i\le n-1$ there is a distance that occurs exactly $i$ times."

Since $1+2+\cdots+(n-1)=\binom n2$, the multiplicities account for every pair of points (p. 1); the paper explains the name by the increasing multiplicities (p. 2).

Proof pointer

A definition; nothing to prove. Figure 1 (p. 2) gives the coordinates of Palásti's eight-point planar example, $(0,1)$, $(\sqrt3,0)$, $(2\sqrt3,0)$, $(\tfrac{5\sqrt3}2,\tfrac52)$, $(\tfrac{3\sqrt3}2,\tfrac92)$, $(\tfrac{\sqrt3}2,\tfrac72)$, $(\tfrac{3\sqrt3}2,\tfrac72)$, $(\sqrt3,2)$, attributed to the paper's reference [Pal89].

Dependencies

None.

Bears on

• Problem 217: for $d=2$, general position is the problem's "no three on a line and no four on a circle", and the multiplicity condition is the problem's requirement that the $n-1$ distinct distances can be ordered so that the $i$th occurs $i$ times; the problem asks for which $n$ a planar crescent configuration of $n$ points exists. The definition decides no instance.