We study facility location in multidimensional normed spaces with the objective of minimizing social cost, the sum of agents' distances to their nearest facility. A deterministic mechanism selects $k$ facility locations based on the reports of $n$ agents and is strategyproof (SP) if no agent can reduce her distance by misreporting her location. We establish optimal approximation ratios and impossibility results. For a single facility in the Euclidean plane, we prove that the optimal approximation ratios among deterministic SP mechanisms are exactly those obtained by Goel and Hann-Caruthers [SCW 2023] for the coordinate-wise median (CM): $\frac{\sqrt{2(n^2+1)}}{n+1}$ for odd $n\ge3$ and $\sqrt{2}$ for even $n\ge4$. We remove the anonymity assumption from their optimality result. However, CM is not always optimal in non-Euclidean spaces: we give an SP mechanism that minimizes social cost exactly in $\ell_\infty^d$, whereas CM has a tight uniform approximation ratio of $2$ already in the plane. We then consider two facilities in $\ell_p$ spaces with $1<p<\infty$ and prove two impossibility results for every $n>2$. In the plane, no deterministic SP mechanism has a finite approximation ratio. In higher dimensions, the same impossibility holds for anonymous deterministic SP mechanisms. Finally, a reduction using distant agents and a limiting argument extends both impossibility results to every $k>2$ with $n>k$.

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