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Source. Problem 7, p. 6 (Section 3, "Restricted point sets in $\mathbb R^2$", pp. 4--6), with Table 1 (p. 4), of Adam Sheffer, Distinct Distances: Open Problems and Current Bounds, arXiv:1406.1949v3 (2 July 2018), the edition read for the source card.

Statement

Notation (pp. 5--6). $D_{\mathrm{conv}}(n)$ is the least number of distinct distances determined by $n$ points in (strict) convex position in the plane. $\hat D_{\mathrm{conv}}(n)$ is the largest number such that every set $\mathcal P$ of $n$ points in convex position has a point $p$ with at least $\hat D_{\mathrm{conv}}(n)$ distinct distances to the points of $\mathcal P\setminus\{p\}$.

What the survey records (p. 6), none of it proved in the survey beyond the remark on Lemma 3.1:

• The regular $n$-gon gives $D_{\mathrm{conv}}(n)\le\lfloor n/2\rfloor$ and $\hat D_{\mathrm{conv}}(n)\le\lfloor n/2\rfloor$.
• Erdős conjectured $D_{\mathrm{conv}}(n)=\lfloor n/2\rfloor$ in 1946, and Altman proved it; Erdős then conjectured $\hat D_{\mathrm{conv}}(n)=\lfloor n/2\rfloor$.
• Lower bounds for $\hat D_{\mathrm{conv}}(n)$: $\lceil(n-1)/3\rceil$ from Lemma 3.1, since convex position has no three collinear points; $\lceil(13n-6)/36\rceil$ (Dumitrescu, 2006); and $\left(\frac{13}{36}+\frac{1}{22701}\right)n+O(1)$ (Nivasch, Pach, Pinchasi and Zerbib), the bound Table 1 lists.

Problem 7 (p. 6). "Find the exact value of $\hat D_{\mathrm{conv}}(n)$." (quoted)

Read depth

Claims checked on the print. The cited results are reported as the survey states them and were not checked against their sources here.

Bears on

• Problem 982: the problem asks whether some vertex of every convex $n$-gon has at least $\lfloor n/2\rfloor$ distinct distances to the other vertices, that is, Erdős's conjecture $\hat D_{\mathrm{conv}}(n)=\lfloor n/2\rfloor$ in the survey's notation, the regular $n$-gon giving the upper bound. The survey records the conjecture open as Problem 7, with the lower bound $\left(\frac{13}{36}+\frac{1}{22701}\right)n+O(1)$.
• Problem 93: the problem's statement is $D_{\mathrm{conv}}(n)\ge\lfloor n/2\rfloor$, which the survey records (p. 6) as proved by Altman; the survey states this and does not prove it.