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Claim. Every set of $n$ points in convex position in the plane has a point from which there are at least $\lceil n/3\rceil$ distinct distances to the other points. This is the theorem as Dumitrescu restates it, "Theorem 1 (Moser)", citing L. Moser, On different distances determined by $n$ points, Amer. Math. Monthly 59 (1952), 85--91 (Dumitrescu 2006, p. 503). Erdős and Fishburn give the same bound as $f(n)\ge\lfloor(n+2)/3\rfloor$ and note that Moser applied his bound to the total number of distances among the vertices, while others, Altman among them, observed that his proof gives the bound at a single vertex (Erdős and Fishburn 1994, pp. 111--112). The argument takes the smallest disk enclosing the polygon and finds a vertex on its boundary whose distances to the vertices of one cap cut off by a chord are all distinct.

Covers. The statement of Problem 982 for $n=3,4,5,7$, where $\lceil n/3\rceil=\lfloor n/2\rfloor$. For every other $n\ge6$ the bound is smaller than $\lfloor n/2\rfloor$.

Depends on. Nothing in this wiki; the result rests on the cited paper. The paper is not filed in the library, and the statement above is taken from the two restatements cited.

Dating. The paper appeared in volume 59, number 2, the February 1952 issue (Crossref record); the day in the page name is a placeholder.

Acceptance. Refereed: The American Mathematical Monthly 59 (1952), no. 2, 85--91. The site's commentary lists the bound; its label, FALSIFIABLE, settles nothing, so the curator's listing is not counted as review.