Metric $1$-median selection: Query complexity vs. approximation ratio

Consider the problem of finding a point in a metric space $(\{1,2,\dots ,n\},d)$ with the minimum average distance to other points. We show that this problem has no deterministic $o(n^{1+1\mathrm{/}(h-1)})$-query $(2h-\mathrm{\Omega }(1))$-approximation algorithms for any constant $h\in {\mathbb{Z}}^{+}\setminus \{1\}$.