Erdős Problem #1131

The problem

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

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

Statement quoted from erdosproblems.com/1131, maintained by Thomas Bloom. Original source: [Er61,p.67][ESVV94][Er95e][Va99,2.45]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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