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Source. Lemma 1, p. 69, 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. The paper introduces Lemmas 1--3 as lemmas from Altman, its reference [1] (Amer. Math. Monthly 70 (1963), 148--157), and does not prove them.

Read depth. Claims checked: the definition of a max side and the statement were read clause by clause on the page image. Nothing here is independently reviewed.

Statement

Definition (p. 69). In a convex polygon whose distinct intervertex distances are $d_1>d_2>\cdots$, a side $xy$ is "max" if $xy=d_1$, and "uniquely max" if $xy=d_1$ and no other side or diagonal has length $d_1$. $m(C)$ is the number of distinct intervertex distances of $C$.

Lemma 1 (p. 69, quoted). "If a side of convex $n$-gon $C$ is max, then $m(C)\geqslant n-2$. If a side of $C$ is uniquely max, then $m(C)\geqslant n-1$."

Lemmas 2 and 3 (p. 69, also from Altman) describe, in the equality cases $m(C)=n-1$ with a uniquely max side $(1,n)$ and $m(C)=n-2$ with a max side $(1,n)$, the vertices labelled $1,\ldots,n$ around the perimeter, which of $d_1,d_2,\ldots$ each chord $(k,n-k+1)$, $(k,n-k+2)$ and $(k-1,n-k+1)$ carries (Fig. 3, p. 70).

Proof pointer

Not proved in this paper; cited from Altman [1]. The paper applies the lemma to subpolygons of consecutive vertices that have a longest segment as a side, in Sections 2--5 (for example Lemma 4, p. 72).

Bears on

• Problem 132: a tool of the convex-position analysis behind Theorem 2; the lemma itself counts distinct distances and says nothing about their multiplicities.