Cambie 2026 maximum product distances diameter point sets

../

---

Stijn Cambie, Arne Decadt, Yanni Dong, Tao Hu, Quanyu Tang, On the maximum product of distances of diameter $2$ point sets. arXiv:2603.07088 (2026).

For a configuration $P=\{z_1,\ldots,z_n\}\subset\mathbb C$ of diameter at most $2$, the paper writes

$$\Delta(P)=\prod_{i\ne j}|z_i-z_j| \quad\text{and}\quad \overline\Delta(P)=\frac{\Delta(P)}{n^n}.$$

The regular diameter-$2$ polygon has $\overline\Delta=1$ for even $n$ and $\overline\Delta\sim e^{\pi^2/8}$ for odd $n$ (Section 1.2). The paper gives structural necessary conditions for a maximizer, exact results at the smallest orders, and two different asymptotic constructions that beat the regular polygon at even orders. It does not determine the maximum in general.

Source. arXiv:2603.07088. The arXiv record (https://arxiv.org/abs/2603.07088, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Markdown. A complete reading copy sits beside the PDF.

Read status. Claims checked against the complete Markdown reading copy. The proofs of Proposition 7 and Theorems 10 and 12, and the proof mechanisms for Theorems 2 and 3, were followed in that copy. The computer-assisted small-order searches, Maple integral evaluations, and the source PDF have not been independently verified.

Bears on. #1045

Conjectural target

Conjecture 4 (Section 1.5) predicts that every odd-order maximizer is the regular $n$-gon; every even-order maximizer has an axis of symmetry, with $2\pi/3$ rotational symmetry when $6\mid n$; and the even-order diameter graph is obtained from $C_{n-3}$ by attaching three pendant edges to cycle vertices. The paper identifies Conjecture 4(i), or even the universal estimate $\overline\Delta(P)\le e^{\pi^2/8}$, as the main surviving challenge. Thus its even-order counterconstructions do not settle the odd-order part of E1045.

Structure of maximizers

• Proposition 7 (Section 2, after Lemma 6) says that every point of a global maximizer is an extreme point of its convex hull. Lemma 6 first uses the maximum principle for the harmonic function $z\mapsto\sum_{j\ne k}\log|z-z_j|$ to show that every vertex is incident to a diameter. A diametral pair of the convex hull consists of exposed points, so the configuration is a convex $n$-gon.
• Theorem 10 (Section 2, after Lemma 9) gives the more precise diameter-graph alternative: it is a caterpillar, or it consists of an odd cycle with extra vertices joined to cycle vertices, every edge being incident with the odd cycle. Proposition 7 and Lemma 9 make the straight-line diameter graph a thrackle; Woodall's classification supplies the graph types, and Lemma 8's harmonic translation argument supplies connectedness. The paper presents Theorem 10 as the more precise version of Theorem 1 (caterpillar or unicyclic). Lemma 11 separately excludes even cycles.
• Theorem 12 (Section 2, equation (1)) applies the Mangasarian--Fromovitz constraint qualification to every local maximizer of the logarithmic nonlinear program. There are multipliers $\lambda_{j,k}\ge0$ with $\lambda_{j,k}(|z_k-z_j|^2-4)=0$ and, for every $k$,

$$\sum_{j\ne k}\frac1{z_j-z_k} =\sum_{j<k}\lambda_{j,k}(\overline z_j-\overline z_k) +\sum_{j>k}\lambda_{k,j}(\overline z_j-\overline z_k).$$

The inward radial direction verifies the constraint qualification. By complementary slackness, only diameter edges can carry nonzero multipliers; equation (1) is therefore a finite equilibrium system indexed by the active diameter graph. Appendix A uses it to prove the $n=4$ classification.

Small orders

• Proposition 13 (Section 3.1) proves $\overline\Delta_{\max}(0)=\overline\Delta_{\max}(1) =\overline\Delta_{\max}(2)=1$ and $\overline\Delta_{\max}(3)=64/27$.
• Proposition 14 (Section 3.1; full proof in Appendix A, equations (15)--(28)) proves $\overline\Delta_{\max}(4)=16(7-4\sqrt3)\approx1.148748$ and classifies the maximizer, up to congruence and relabeling, as the kite $\{0,2,\sqrt3+i,\sqrt3-i\}$.
• Proposition 15 (Section 3.2) proves $\overline\Delta_{\max}(5)=(4/5)^5(\sqrt5-1)^{10}$, uniquely at the regular pentagon. Following its numerical-search discussion, Section 3.2 reports $\overline\Delta_{\max}(6)=(2\sqrt3-2)^{18}/3^6\approx1.310854$.
• Beyond Propositions 13--15, Sections 3.2--3.5 do not prove the displayed configurations globally optimal. They report search evidence for the regular odd polygons through $n=11$ (Section 3.2), conjectural optima for $n=8$ and $10$, a $D_3$-symmetric $n=12$ candidate with value approximately $1.2901383629$, and lower bounds found numerically for even $4\le n\le68$ (Table 1). This distinction matters: the labeled global proofs in this section cover $n\le5$; the $n=6$ value is stated after the authors' computational search rather than supplied with a comparable standalone proof.

Even-order constructions

Theorem 2 (Section 1.4; proof in Section 4, Lemmas 16--19) constructs, for $6\mid n$, a diameter-$2$ polygon $P$ such that

$$\overline\Delta(P)\longrightarrow C_*=\frac{3^{9/4}}{2^3} \exp\!\left(\frac{\pi^2-2\sqrt3\,\pi}{8}\right) \approx1.304457.$$

It also proves $\liminf_{n\to\infty}\overline\Delta_{\max}(n)\ge C_*^{1/9}>1$. The construction starts with six congruent arcs of a regular $6k$-gon, changes the six junction angles alternately by $\pm\pi/n$, and rescales by $2/\cos(\pi/2n)$. Lemma 16 controls the diameter. Lemmas 17--19 split the distance-product ratio into three regimes with limits $C_1,C_2,C_3$; combining them with the rescaling factor gives $C_*$. Taking every third vertex in a $3n$-vertex construction explains the ninth root in the all-order liminf.

Theorem 3 (Section 1.4; proof in Section 5) proves the uniform even-order bound

$$\liminf_{\substack{n\to\infty\\n\ \mathrm{even}}} \overline\Delta_{\max}(n) \ge \exp\!\left(\frac7{24}\zeta(3)-\frac{\pi^4}{864}\right) \approx1.26853.$$

Here $z_k=(1+t_ng(2\pi k/n))e^{2\pi ik/n}$ with $g(\theta)=\operatorname{tri}(3\theta)$ and $t_n=\frac{\pi^2}{12n}(1-1/n)$. The $\pi$-antiperiodicity of $g$ keeps antipodal distances equal to $2$; its Lipschitz bound controls every other pair (Lemma 21, equations (8)--(10)). The factorization in equations (11)--(13) and the same antiperiodicity cancel the linear term (Lemma 25), leaving a quadratic term. Lemmas 27 and 31--35 pass it to a Fourier-evaluated integral $J=1/3-84\zeta(3)/\pi^4<0$, producing the positive limiting exponent.

Historical qualification

The recent paper should not be cited as the first disproof of regular-polygon optimality for even $n$. Erdős, Herzog and Piranian posed the question as Problem 13 in 1958 (printed p. 143). Pommerenke's 1961 paper supplied the general upper bound $\Delta\le 2^{O(n)}n^n$. More importantly, L. Danzer and Ch. Pommerenke, Über die Diskriminante von Mengen gegebenen Durchmessers, Monatshefte für Mathematik 71 (1967), 100--113, had already disproved the regular-polygon conjecture for even $n$. Erdős explicitly records that chronology in Extremal Problems on Polynomials (1976), p. 350, while retaining the odd-$n$ conjecture. The contribution here is the new structural theory, small-order analysis, explicit modern families, and quantitative asymptotic lower bounds, not the first historical counterexample.