We study strategyproof mechanisms for eliciting agents' location preferences truthfully in the Euclidean plane $\mathbb R^2$ and locating a facility so as to minimize the $L_p$-norm social cost, defined as the $L_p$-norm of the vector of distances from the facility to the agents' preferred locations, for any $p \ge 1$. While the cases $p=1$ and $p=\infty$ have been well-studied, open questions remain about the optimal approximation ratios achievable by strategyproof mechanisms for general $p$. Our first result resolves an open question of Goel and Hann-Caruthers [Soc. Choice Welf. 2023]. They showed that the coordinate-wise median (CM) mechanism achieves an approximation ratio lying between \(2^{1-\frac{1}{p}}\) and \(2^{\frac{3}{2}-\frac{2}{p}}\) for $p\ge 2$, and they conjectured that it is exactly \(2^{1-\frac{1}{p}}\). We confirm this conjecture, and we further show that CM has a tight $\sqrt 2$-approximation for $1\le p\le 2$. Since it is previously known that the CM mechanism has the optimal approximation ratio among all deterministic anonymous strategyproof mechanisms for all $p\ge 1$, we complete the picture of deterministic mechanisms. Our second and third results demonstrate that two randomized mechanisms can yield better approximation ratios. In particular, we first consider the uniformly rotated coordinate-wise median (URCM) mechanism, and prove that, for \(1\le p<2\), its approximation ratio strictly improves over the deterministic bound \(\sqrt{2}\), while no such improvement is possible for $p\ge 2$. We then study the centroid random dictatorship mechanism that returns the average location (i.e., centroid) and the random dictatorship each with half probability, and show that its approximation ratio strictly improves over CM and URCM for every finite \(p\gtrsim 1.6\).

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