Problem 1131

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}?

Source: erdosproblems.com/1131

Analysis
Polynomials

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Status

Open

No claim settles this problem.

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.

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.

Problem 1131

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