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Sparse perturbation

MathOverflow Asked by Yiming Xu on December 1, 2020

Let $x, x_0inmathbb{R}^n$ be two vectors satisfying $$frac{|x|_1}{|x|_2}leqfrac{|x_0|_1}{|x_0|_2}.$$
$| cdot|_1$ and $| cdot|_2$ are the $ell_1$ and $ell_2$ norm in $mathbb{R}^n$, respectively. Suppose that $$|x_0|_0:= |{1leq ileq n: (x_0)_ineq 0}|=s.$$

The questions is: Does there exist a constant $c$ independent of $n$ and $s$ such that
$$frac{|x-x_0|_1}{|x-x_0|_2}leq csqrt{s}?$$

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