  We consider the classic 1-center problem: Given a set $P$ of $n$ points in a
metric space find the point in $P$ that minimizes the maximum distance to the
other points of $P$. We study the complexity of this problem in $d$-dimensional
$\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our
results for the 1-center problem may be classified based on $d$ as follows.
  $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that
when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem
in any of the $\ell_p$-metrics, or in edit or Ulam metrics.
  $\bullet$ Large $d$: When $d=\Omega(n)$, we extend our conditional lower
bound to rule out subquartic algorithms for 1-center problem in edit metric
(assuming Quantified SETH). On the other hand, we give a
$(1+\epsilon)$-approximation for 1-center in Ulam metric with running time
$\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$.
  We also strengthen some of the above lower bounds by allowing approximations
or by reducing the dimension $d$, but only against a weaker class of algorithms
which list all requisite solutions. Moreover, we extend one of our hardness
results to rule out subquartic algorithms for the well-studied 1-median problem
in the edit metric, where given a set of $n$ strings each of length $n$, the
goal is to find a string in the set that minimizes the sum of the edit
distances to the rest of the strings in the set.
