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Updated Oct 8

Problem 982

claims/: The 3 claim pages of Problem 982, one per claimant's result; the problem's standing derives from them.

Statement. If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor \frac{n}{2}\rfloor$ different distances to other vertices.

Status. Falsifiable: the site labels the problem FALSIFIABLE, a counterexample being a finite check, and its page was last edited on 19 October 2025. Three refereed lower bounds the site credits settle small cases and have accepted partial claim pages: Moser 1952, Erdős and Fishburn 1994 and Dumitrescu 2006. The proof-claims tab carries one partial claim, a lower bound submitted 2026-07-25 by Scott Duke Kominers, which has no claim page for the reason given under Current assessment; the label was unchanged on 2026-10-06.

Source. erdosproblems.com/982, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #982, https://www.erdosproblems.com/982.

References.

[BaRo13] Bárány, Imre and Roldán-Pensado, Edgardo, A question from a famous paper of Erdős. Discrete Comput. Geom. (2013), 253-261.
[Du06b] Dumitrescu, Adrian, On distinct distances from a vertex of a convex polygon. Discrete Comput. Geom. 36 (2006), 503-509.
[Er46b] Erdős, P., On sets of distances of $n$ points. Amer. Math. Monthly (1946), 248-250.
[ErFi94] Erdős, Paul and Fishburn, Peter, A postscript on distances in convex $n$ -gons. Discrete Comput. Geom. (1994), 111-117.
[Mo52] Moser, Leo, On the different distances determined by $n$ points. Amer. Math. Monthly (1952), 85-91.
[NPPZ13] Nivasch, Gabriel and Pach, János and Pinchasi, Rom and Zerbib, Shira, The number of distinct distances from a vertex of a convex polygon. J. Comput. Geom. (2013), 1-12.

Formalization. Statement in formal-conjectures.

Current assessment

The claim pages record the cases the published lower bounds settle: the statement holds for every $n\le7$ and for $n=9$ , $n=8$ is the smallest open case, and the problem is open. The statement is falsifiable: a convex polygon in which every vertex sees fewer than $\lfloor n/2\rfloor$ distinct distances would refute it by a finite computation, and none is known. The regular polygon shows that $\lfloor n/2\rfloor$ would be sharp.

Forum claim without a page. The site's proof-claims tab carries a partial claim by Scott Duke Kominers, posted under the forum account skominers on 2026-07-25 (tab entry), with a write-up and no formalization; the entry names GPT 5.6 Sol and Claude Fable 5 as the systems used. Writing $f(n)$ for the largest number such that every convex $n$ -gon has a vertex with at least $f(n)$ distinct distances to the other vertices, it claims

f(n)\ge\Bigl(\frac{13}{36}+\frac{3}{5270}\Bigr)n-O(1),

which improves the additive term $1/22701$ in the coefficient of Nivasch, Pach, Pinchasi and Zerbib [NPPZ13] by a factor of about $12.9$ and leaves the leading term $13/36$ of Dumitrescu [Du06b] unchanged, by refining their iteration over good edges and witnesses. The claim has no page because it is a better lower bound on a falsifiable statement that settles no case of it: it proves the $\lfloor n/2\rfloor$ bound for no $n$ and for no class of polygons, and its author calls it a long way from the conjecture. No review of the write-up is recorded, and the site's page, last edited before the claim, does not mention it.

Known results. The site records the lower bounds $f(n)\ge\lceil n/3\rceil$ of Moser [Mo52], $f(n)\ge\lfloor n/3+1\rfloor$ of Erdős and Fishburn [ErFi94], $f(n)\ge\lceil(13n-6)/36\rceil$ of Dumitrescu [Du06b] and $f(n)\ge(13/36+1/22701)n-O(1)$ of Nivasch, Pach, Pinchasi and Zerbib [NPPZ13], and the counterexamples of Bárány and Roldán-Pensado [BaRo13] to Erdős's stronger conjecture about convex curves. The first three bounds have the claim pages named under Status; the last two results follow the site's remarks, and the bound of [NPPZ13], with its unspecified $O(1)$ , settles no case and has no claim page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (30)

Problem 93 Moser's bound, a vertex with at least n/3 distinct distances Erdős and Fishburn, a run of n/3 + 1 successively farther vertices Dumitrescu, a vertex with at least (13n-6)/36 distinct distances problems/distance_problems/E0982/claims Problem 1082 Altman's convex polygon bound settles convex position Distance Problems distance_problems/altman_1963_problem_p_erdos distance_problems/barany_2013_question_famous_paper_erdos Theorem 1.1: there is a planar convex body K with N(K) = 6 Theorem 1.2: N(K) < ∞ for every planar convex body K Theorem 1.3: convex bodies with |J(K_ε, ∞)| > (1 − ε)|∂K_ε|Theorem 1.4: for most convex bodies, most boundary points lie in every J(K, n)distance_problems/dumitrescu_2006_distinct_distances_vertex_convex_polygon Theorem 2 (p. 504): a vertex with at least (13n-6)/36 distinct distances distance_problems/erdos_1946_sets_distances_points Conjectures (p. 248): a convex n-gon determines at least [n/2] distances, and has a vertex with no three vertices equidistant from it distance_problems/erdos_1975_problems_elementary_combinatorial_geometry Section 1, the three convex-polygon conjectures (p. 100): Altman's theorem, the single-vertex form, and Danzer's counterexample distance_problems/erdos_1994_postscript_distances_convex_gons Inequality (p. 116): for n ≥ 4 some vertex of a convex n-gon has at least ⌊(n+3)/3⌋ distinct distances Theorem (p. 112): for n ≥ 4 the guaranteed longest run in a convex n-gon is ⌊(n+3)/3⌋Erdős–Fishburn: Multiplicities of interpoint distances in finite planar sets distance_problems/nivasch_2013_number_distinct_distances_vertex_convex_polygon Lemma 2 (p. 3): isosceles-triangle bounds give distinct distances Theorem 1 (p. 2): a vertex with (13/36 + ε)n - O(1) distinct distances Theorem 9 (p. 6): at least n²/11.981 good edges in convex position distance_problems/sheffer_2014_distinct_distances_open_problems_current_bounds Problem 7 (p. 6): the exact value of the single-point distance count for convex position

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