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Statement

Section 2 (p. 248) poses three conjectures, each stated by the paper as stronger than the one before. None is proved in the paper; the paper says it is unable to prove the first.

1. Distinct distances in convex position. If the $n$ points form a convex polygon, then $f(n)\ge[n/2]$, where $f(n)$ is now the least number of distinct distances among such points, with equality when the points are the vertices of a regular $n$-gon.
2. An equidistance-free vertex (quoted). "In every convex polygon there is at least one vertex with the property that no three vertices of the polygon are equally distant from it." The paper notes that the distances from such a vertex would then give $[n/2]$ different distances.
3. Convex curves. On every convex curve there is a point $P$ such that every circle with centre $P$ meets the curve in at most $2$ points.

Source. P. Erdős, On sets of distances of $n$ points, Amer. Math. Monthly 53 (1946), 248--250; Section 2, on p. 248. The copy read is identified on the source card.

Read depth. Claims checked: the three statements were read clause by clause on the page image. Nothing here is independently reviewed.

Proof pointer

No proof; these are conjectures. The paper's one argument is the step from the second to the first: a vertex from which no three vertices are equally distant sees each distance at most twice among the other $n-1$ vertices, so it has at least $[n/2]$ distinct distances to them.

Dependencies

None.

Bears on

• Problem 93: the problem's statement, that $n$ points forming a convex polygon determine at least $\lfloor n/2\rfloor$ distinct distances, is the first conjecture without its equality clause.
• Problem 982: the problem asks for a vertex with at least $\lfloor n/2\rfloor$ distinct distances to the other vertices, the consequence the paper draws from the second conjecture; the paper does not pose it as a separate conjecture.
• Problem 97: the problem asks for a vertex with no four other vertices equidistant from it, a weaker form of the second conjecture, which has three; the problem page records Danzer's convex nonagon against the three-vertex form.