Problem 1153

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 $i\neq k$ .

Let

\lambda(x)=\sum_k \lvert l_k(x)\rvert.

Is it true that, for any fixed $-1\leq a< b\leq 1$ ,

\max_{x\in [a,b]}\lambda(x)> \left(\frac{2}{\pi}-o(1)\right)\log n?

Source: erdosproblems.com/1153

Analysis
Polynomials

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. The catalog labels the problem PROVED (page last edited 1 April 2026, accessed 2026-10-07), crediting Tao [Ta26b]; the frontmatter standing (solved, proved) derives from the accepted claim page Tao 2026. The status-defining source is Tao's arXiv:2603.21453v3 preprint, Theorem 1.10(i), with the elementary transfer below; no journal acceptance, compilation of the complete source proof or independent review of that proof is recorded, and the site's proof-claims page lists one unexamined alternative proof, the pending claim page Yang 2026 (Formalization). The refereed whole-interval bound of Erdős, the case $a=-1$ , $b=1$ , has the accepted partial claim page Erdős 1961.

Be31: S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degré n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.
Er61c: Erdős, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.
ErSz78: Erdős, P. and Szabados, J., On the integral of the Lebesgue function of interpolation. Acta Math. Acad. Sci. Hungar. 32 (1--2) (1978), 191--195. The site's reference record resolves the key [ErSz78] to Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients, Austral. Math. Soc. Gaz. (1978), 97-99, which does not concern this problem; the entry above is the paper the commentary describes, whose integral bound gives the maximum bound $\max_{[a,b]}\lambda\gg\log n$ that the commentary states.
ErTu61: Erdős, P. and Turán, P., An extremal problem in the theory of interpolation. Acta Math. Acad. Sci. Hungar. (1961), 221-234.
Ta26b: T. Tao, Local Bernstein theory, and lower bounds for Lebesgue constants. arXiv:2603.21453v3 (2026).
Va99: Various contributors, Some of Paul’s favorite problems. July 1999 booklet, item 2.44, on the right leaf of PDF p. 5 of Kovač's public scan.

Problem 1153

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Proof claims

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