---

Claim. A run of a convex $n$-gon from a vertex $x_0$ is a sequence $x_0,x_1,\ldots,x_k$ of successively adjacent vertices, going clockwise or counterclockwise, with $d(x_0,x_1)<d(x_0,x_2)<\cdots<d(x_0,x_k)$; its length is $k$, and $g(n)$ is the minimum over convex $n$-gons of the longest run. The Theorem of P. Erdős and P. Fishburn, A postscript on distances in convex $n$-gons, Discrete Comput. Geom. 11 (1994), 111--117 (p. 112), reads: "For all $n \geq 4$, $g(n) = \lfloor (n + 3)/3 \rfloor$." The lower bound extends Moser's 1952 argument, and an explicit example gives the upper bound. A run of length $k$ from $x_0$ gives $k$ distinct distances from $x_0$, so every convex $n$-gon with $n\ge4$ has a vertex with at least $\lfloor n/3\rfloor+1$ distinct distances to the other vertices. The paper states this consequence itself, as the inequality $f(n)\ge\lfloor(n+3)/3\rfloor$ for $n\ge4$ on p. 116, and calls it a tiny improvement on Moser's $\lfloor(n+2)/3\rfloor$; it is the bound the site's commentary credits to the paper. The paper's introduction (p. 111) names Moser's bound as the best previously known lower bound on the number of distances at one vertex.

Covers. The statement of Problem 982 for $n=4,5,6,7,9$, where $\lfloor n/3\rfloor+1=\lfloor n/2\rfloor$. For $n=8$ and every $n\ge10$ the bound is smaller than $\lfloor n/2\rfloor$.

Depends on. Nothing in this wiki; the result rests on the cited paper.

Dating. The paper appeared in volume 11, number 1, the January 1994 issue (Crossref record); the day in the page name is a placeholder.

Source card. erdos_1994_postscript_distances_convex_gons.

Acceptance. Refereed: Discrete & Computational Geometry 11 (1994), no. 1, 111--117. The site's commentary credits the bound; its label, FALSIFIABLE, settles nothing, so the curator's credit is not counted as review.