Problem 1131

Erdős Problem #1131: declared status 'open'. Formalized: no. 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$$\min I =2-(1+o(1))\frac{1}{n}?$$ Current best: Erd\H{o}s, Szabados, Varma, and V\'{e}rtesi [ESVV94] proved that$$2-O\left(\frac{(\log n)^2}{n}\right)\leq \min I\leq 2-\frac{2}{2n-1}$$where the upper bound is witnessed by the roots of the integral of the Legendre polynomial as above. Prize: no. Tags: analysis, polynomials.