---

Claim. The answer to Problem 957 is yes. Adrian Dumitrescu, A product inequality for extreme distances, in 35th International Symposium on Computational Geometry (SoCG 2019), LIPIcs 129, 30:1--30:12, published 11 June 2019, and in Comput. Geom. 85 (2019), 101577. Theorem 1 states that for $n$ distinct points in the plane, if the minimum distance occurs $s_{\min}$ times and the maximum distance occurs $s_{\max}$ times, then

$$s_{\min}\,s_{\max}\leq\tfrac{9}{8}n^2+O(n).$$

In the problem's notation $s_{\min}=f(d_1)$ and $s_{\max}=f(d_k)$, so the inequality asked for holds with a linear error term in place of $o(n^2)$. The classical bounds $f(d_1)\leq3n$ and $f(d_k)\leq n$ give only $3n^2$. The constant is sharp: the paper's Figure 1 exhibits $3n/4$ hull points, the center of a circular arc subtending $60^\circ$ together with $3n/4-1$ points spaced at unit distance along the arc, whose radius is the diameter of the set, and $n/4$ interior points forming a piece of the unit triangular lattice. The center is at the diameter from every arc point and the $60^\circ$ chord between the ends of the arc is a further diameter, so $f(d_k)=\frac{3}{4}n$, while the arc and the lattice give $f(d_1)=\frac{3}{2}n-O(\sqrt n)$, so the product is $\frac{9}{8}n^2-O(n\sqrt n)$; Erdős and Pach had attributed such a construction to Makai. The proof works in the minimum-distance graph and the diameter graph: it splits the points into hull vertices, other boundary points and interior points, uses that all but $O(1)$ hull vertices have flat neighborhoods (seven consecutive interior angles in $(179^\circ,180^\circ)$), and combines the edge bounds of the two graphs with the degree bound $6$ in the minimum-distance graph. The paper is carded at dumitrescu_2019_product_inequality_extreme_distances.

Acceptance. The result is refereed: it appeared in Computational Geometry, after the SoCG 2019 proceedings. The site's curator, Thomas Bloom, marks the problem PROVED and credits Dumitrescu's theorem on the problem page; the curator neither wrote nor submitted the result. This corpus has not reviewed the proof, and no such review is needed for the standing recorded here. The site's further remarks, the sum bound $f(d_1)+f(d_k)\leq3n-c\sqrt n+o(\sqrt n)$ with its best constant, and the stronger conjecture $f(d_1)\leq3n-2m+o(\sqrt n)$ for a hull of $m$ vertices, are not part of the question and are not settled by this claim.