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Claim. Theorem 2 of A. Dumitrescu, On distinct distances from a vertex of a convex polygon, Discrete Comput. Geom. 36 (2006), 503--509 (p. 504): "Let $P$ be a set of $n$ points in convex position in the plane. Then there exists a point $p \in P$ such that the number of distinct distances from $p$ is at least $\lceil (13n - 6)/36 \rceil$." The proof starts, as Moser's does, from the smallest disk enclosing the points and adds a count of the isosceles triangles the points determine.

Covers. The statement of Problem 982 for $n=3,4,5,7,9$, where $\lceil(13n-6)/36\rceil=\lfloor n/2\rfloor$. For $n=6$, $n=8$ 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. Received 30 June 2005 and published online 29 September 2006, the date this page is named by; the print issue is volume 36, number 4 (December 2006).

Source card. dumitrescu_2006_distinct_distances_vertex_convex_polygon.

Acceptance. Refereed: Discrete & Computational Geometry 36 (2006), no. 4, 503--509. The site's commentary credits the bound; its label, FALSIFIABLE, settles nothing, so the curator's credit is not counted as review.