Problem 1131

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claims/: The 1 claim page of Problem 1131, one per claimant's result; the problem's standing derives from them.

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Statement. For $x_1,\ldots,x_n\in [-1,1]$ let

$$l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},$$

which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.

What is the minimal value of

$$I(x_1,\ldots,x_n)=\int_{-1}^1 \sum_k \lvert l_k(x)\rvert^2\mathrm{d}x?$$

In particular, is it true that

$$\min I =2-(1+o(1))\frac{1}{n}?$$

Status. Open. The site labels the problem OPEN (page last edited 2026-01-23). Its discussion thread carries one pending partial claim, recorded and not adopted: Price 2026, posted on 2026-04-26, which credits GPT-5.5 Pro with a disproof of the displayed asymptotic $\min I=2-(1+o(1))/n$; the minimal value of $I$ is not determined by it. The thread's other postings are recorded under Current assessment with the reasons they are not claims.

Source. erdosproblems.com/1131, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1131, https://www.erdosproblems.com/1131.

References.

• [ESVV94] Erdős, P. and Szabados, J. and Varma, A. K. and Vértesi, P., On an interpolation theoretical extremal problem. Studia Sci. Math. Hungar. (1994), 55-60.
• [Fe32] Fejér, Leopold, Bestimmung derjenigen Abszissen eines Intervalles, für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein Möglichst kleines Maximum Besitzt. Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) 1(3) (1932), 263-276.
• [Sz66] Szabados, J., On a problem of P. Erdős. Acta Math. Acad. Sci. Hungar. (1966), 155-157.
• [BrTo97] Brutman, L. and Toledano, D., An extremal problem of Erdős in interpolation theory. Comput. Math. Appl. 34 (1997), no. 12, 37-47.

Formalization. None recorded: the community database lists the problem as unformalized and no formal-conjectures statement file exists for it.

Current assessment

The question (site formulation, page last edited 2026-01-23). For $n$ nodes in $[-1,1]$ with Lagrange basis polynomials $l_k$, the least value of $I=\int_{-1}^1\sum_k|l_k(x)|^2\,dx$, and whether $\min I=2-(1+o(1))/n$. The site labels the problem OPEN. Erdős at first conjectured that the minimum is attained at the roots of the integral of the Legendre polynomial, the nodes Fejér [Fe32] had shown to minimize $\max_{x\in[-1,1]}\sum_k|l_k(x)|^2$; Szabados [Sz66] disproved that conjecture for every $n>3$. Erdős, Szabados, Varma and Vértesi [ESVV94] proved

$$2-O\left(\frac{(\log n)^2}{n}\right)\le\min I\le 2-\frac{2}{2n-1},$$

the upper bound attained at the roots of the integral of the Legendre polynomial.

Standing. Open, with one pending partial claim. Price 2026, a write-up posted in the site's discussion thread on 2026-04-26, credits GPT-5.5 Pro with a disproof of the displayed asymptotic; a reply by Nat Sothanaphan on 2026-04-27 reports that a standard check found no issues, and the site's label is unchanged. The claim would answer the second question no and leaves the first open, so the problem derives no settled standing from it. The numerical study of Brutman and Toledano [BrTo97], raised in the thread on 2026-04-07 and again in the reply of 2026-04-27, had earlier pointed against the asymptotic; a thread comment of 2026-08-14 reports its estimate of the limit of $n(2-\min I)$ as about $1.094$. Numerical evidence is not a proof, so [BrTo97] is cited and not paged as a claim.

Rafik's manuscript, not paged as a claim. Zeraoulia Rafik announced in the thread on 2026-01-10 a manuscript, Asymptotics of Erdős's $L^2$ Lagrange interpolation problem: arcsine distribution and Airy endpoint universality (Zenodo records created 2026-01-11 and, revised, 2026-01-18; posted on preprints.org on 2026-01-13), which claims unconditionally that any asymptotically minimizing sequence of nodes equidistributes with respect to the arcsine measure and that the lower bound improves to $\min I\ge 2-O(1/n)$, and, under an endpoint universality conjecture, that $\min I=2-c/n+o(1/n)$ for an explicit constant $c$ given through the Airy kernel, with numerics suggesting $c=2$. The unconditional part is a bound that settles neither question. The conditional part leaves $c$ unevaluated, and the manuscript's own numerical statements conflict, since asymptotic optimality of the Legendre-integral nodes, which it also asserts, would give $c=1$ and not $2$, as the thread comment of 2026-08-14 observes. The manuscript therefore decides neither question even under its hypothesis, and it is recorded here rather than as a claim page. The thread comment of 2026-08-14, by the user mzn, reports certified two-sided brackets for $\min I$ at $n\le5$ and exact rational upper witnesses through $n=10$, computed by branch and bound in exact rational arithmetic with Fable and ChatGPT 5.6 Sol named as checking tools; it confirms Szabados's result with explicit witnesses for $4\le n\le5$ and finds $n(2-\min I)$ decreasing from about $1.333$ at $n=2$ toward the Brutman--Toledano value. It is a thread post and not a dated manuscript, so it gets no page.

Lean coverage. None: the community database lists the problem as unformalized and no formal-conjectures statement file exists for it, so no formalized evidence is available to any claim.

Search scope. The site's problem page (last edited 2026-01-23) and its discussion thread of six comments, as of 2026-10-07; the Zenodo and preprints.org records of Rafik's manuscript and the Crossref record of [BrTo97], as of 2026-10-07; the proof-claims tab, which lists no claim for the problem. Price's write-up is not readable from its link. No proof is compiled in this wiki.

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.

• erdos_1961_problems_results_interpolation_ii
• erdos_1961_problems_results_interpolation_ii / theorem_4
• erdos_1967_problems_results_convergence_divergence_properties_lagrange
• erdos_1967_problems_results_convergence_divergence_properties_lagrange / equation_5
• fejer_1932_bestimmung_derjenigen_abszissen_eines_intervalles
• fejer_1932_bestimmung_derjenigen_abszissen_eines_intervalles / equation_59
• fejer_1932_bestimmung_derjenigen_abszissen_eines_intervalles / equation_69
• fejer_1932_bestimmung_derjenigen_abszissen_eines_intervalles / equation_97
• fejer_1932_bestimmung_derjenigen_abszissen_eines_intervalles / main_theorem