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Denote $S_n=\left\{x_{n}=(x_{n0},x_{n1},...,x_{nn})|a\leq x_{n0}<x_{n1}<\cdots<x_{nn}\leq b\right\}$ with $-\infty<a<b<+\infty$ and $n\geq 1$. For any $x_n\in S_n$, define the Lebesgue function $L_{x_n}(x)$ by $$ L_{x_n}(x):=\sum\limits_{i=0}^{n}\left|\frac{\prod\limits_{0\leq j\leq n,~j\neq i}(x-x_{nj})}{\prod\limits_{0\leq j\leq n,~j\neq i}(x_{ni}-x_{nj})}\right|. $$ How to show that $\inf\limits_{x_n\in S_n}\max\limits_{x\in[a,b]}L_{x_n}(x)\geq C\log n$ for some constant $C>0$ ?

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