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Source. Theorem 1, 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. The paper cites the theorem and does not prove it.

Read depth. Claims checked: the statement, its attribution in the introduction (p. 65) and the notation of p. 66 were read clause by clause on the page images. Nothing here is independently reviewed.

Statement

Setting (p. 66): $\mathscr C_n$ is the class of convex $n$-gons, $m(C)$ the number of distinct distances between vertices of $C$, and $M_n(t)=\{C\in\mathscr C_n:m(C)=t\}$; a class "contains 1 polygon" $R_n$ when it consists exactly of the polygons similar to the regular $n$-gon $R_n$ (see Theorem 2 for the full convention).

Theorem 1 (Altman) (p. 66, quoted). "For every $n\geqslant3$, $m(C)\geqslant\lfloor n/2\rfloor$ for all $C\in\mathscr C_n$. If $n$ is odd then $M_n((n-1)/2)$ contains 1 polygon, $R_n$."

In the corpus's words: the vertices of a convex $n$-gon, $n\ge3$, determine at least $\lfloor n/2\rfloor$ distinct distances, and when $n$ is odd a convex $n$-gon determines exactly $(n-1)/2$ distances if and only if it is regular. The introduction (p. 65) attributes the bound and the odd equality case to Altman's two papers, the paper's references [1] (Amer. Math. Monthly 70 (1963), 148--157) and [2] (Canad. Math. Bull. 15 (1972), 329--340), and the bound to a conjecture of Erdős [3]. Since $m(R_n)=\lfloor n/2\rfloor$ and $m(R_n-k)=\lfloor n/2\rfloor$ when $n>2k$ (p. 66), the bound is attained for every $n$; the even equality cases are Theorem 2.

Proof pointer

Not proved in this paper. The bound is the Theorem of p. 149 of Altman's 1963 paper, recorded on its own page. The odd equality case is cited from Altman's papers jointly, without a locator.

Bears on

• Problem 132: for odd $n$, the only convex $n$-gon with the minimum $(n-1)/2$ distances is $R_n$, in which every distance occurs exactly $n$ times, so all of its distances occur at most $n$ times (a count made here). The statement is for convex position only.
• Problem 93: the first sentence of the theorem is the problem's statement, cited here from Altman; this paper adds no proof of it.