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Source. Theorem 2, p. 66, of Peter Fishburn, "Convex polygons with few intervertex distances," Computational Geometry 5 (1995), no. 2, 65--93, doi:10.1016/0925-7721(94)00020-v, the edition named on the source card.

Read depth. Claims checked: the notation of p. 66, the statement, Fig. 1 (p. 67) and the closing paragraph of Section 3 (p. 81) were read clause by clause on the page images. The proofs of Sections 2 and 3 were read for structure only. Nothing here is independently reviewed.

Statement

Setting (p. 66). For $n\ge3$, $\mathscr C_n$ is the class of all convex $n$-gons in the plane, and $C\approx D$ means that $C$ and $D$ are similar (rotation, reflection, translation and uniform rescaling). A subclass of $\mathscr C_n$ "contains $N$ polygons" when there are pairwise dissimilar $C_1,\ldots,C_N\in\mathscr C_n$ such that the subclass consists of exactly the $C\in\mathscr C_n$ similar to some $C_i$. $R_n$ is the regular $n$-gon and $R_n-k$, for $k\le n-3$, is a regular $n$-gon with $k$ vertices deleted, a member of $\mathscr C_{n-k}$. With vertex set $\{1,\ldots,n\}$, $m(C)=\lvert\{d(i,j):i\ne j\}\rvert$ is the number of distinct intervertex distances of $C$, and $M_n(t)=\{C\in\mathscr C_n: m(C)=t\}$.

Theorem 2 (p. 66, quoted). "$M_4(2)$ contains 4 polygons, and $M_6(3)$ contains 3 polygons: see Fig. 1. For every even $n\geqslant8$, $M_n(n/2)$ contains 2 polygons, $R_n$ and $R_{n+1}-1$."

In the corpus's words: by Altman's bound (Theorem 1) a convex $n$-gon with $n$ even has at least $n/2$ distinct distances, and for every even $n\ge8$ it has exactly $n/2$ if and only if it is similar to $R_n$ or to $R_{n+1}-1$.

The small cases (Fig. 1, p. 67, whose caption reads "$M_{2N}(N)$ is all convex $2N$-gons with exactly $N$ intervertex distances"). Each polygon is labelled with its multiplicity vector, the multiplicities of its distances in decreasing order without regard to which distance has which (pp. 66, 68):

• $M_4(2)$: $A_4$ with $(5,1)$, drawn as a rhombus whose one diagonal is the only other distance; $B_4$ with $(4,2)$; $R_4$ with $(4,2)$; and $R_5-1$ with $(3,3)$.
• $M_6(3)$: $A_6$ with $(6,6,3)$; $R_6$ with $(6,6,3)$; and $R_7-1$ with $(5,5,5)$.
• $M_8(4)$, as an instance of the general case: $R_8$ with $(8,8,8,4)$ and $R_9-1$ with $(7,7,7,7)$.

Proof pointer

The case $M_4(2)$ is stated without proof as straightforward (p. 66). $M_6(3)$ is Section 2 (pp. 69--71), a case analysis on where the longest distance sits, using Altman's Lemma 1 and his Lemmas 2 and 3 (p. 69). The case $n=2N$, $N\ge4$, is Section 3 (pp. 72--81): Lemma 4 (p. 72) confines the longest distances from a chosen vertex 1 to the vertices $N+1$ and possibly $N$ or $N+2$. When $1$ has a single longest segment, Lemma 5 (p. 72) makes every segment between opposite vertices $j$, $j+N$ one of the two longest distances, and further use of Lemmas 1--3 forces equal sides and equal short diagonals on one circle, so $C\approx R_{2N}$ (p. 73). Otherwise Lemmas 6--9 (pp. 73--81) place $N+2$ vertices equally spaced on a circle and then every other vertex on the same circle, giving $C\approx R_{2N+1}-1$. The paper notes (p. 81) that this argument fails at $N=3$, where $A_6$ is a third polygon, and that $N=4$ also follows from the classification of $M_7(4)$ in Theorem 3 by deleting a vertex of an octagon in $M_8(4)$ (p. 72).

Dependencies

Theorem 1 and Lemma 1, with Altman's Lemmas 2 and 3 (p. 69).

Bears on

• Problem 132: the problem asks whether every $n$ points in the plane have two distances that each occur at least once but between at most $n$ pairs. The theorem is stated for the vertices of convex polygons only. Its two polygons for even $n\ge8$ have every distance occurring at most $n$ times: in $R_n$ each distance other than the diameter occurs $n$ times and the diameter $n/2$ times, and in $R_{n+1}-1$ each distance occurs $n-1$ times (a count made here, not printed in the paper). The paper says nothing about sets not in convex position or about the growth of the number of such distances.