The coordinate-wise median is a constant-factor approximation in every L_q(R^d), with UB(2)=sqrt(6*sqrt(3)-8)<1.55 and matching lower bounds as d grows.
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Approximation guarantees of Median Mechanism in $\mathbb{R}^d$
The coordinate-wise median is a constant-factor approximation in every L_q(R^d), with UB(2)=sqrt(6*sqrt(3)-8)<1.55 and matching lower bounds as d grows.