  In the Priority $k$-Center problem, the input consists of a metric space
$(X,d)$, an integer $k$, and for each point $v \in X$ a priority radius $r(v)$.
The goal is to choose $k$-centers $S \subseteq X$ to minimize $\max_{v \in X}
\frac{1}{r(v)} d(v,S)$. If all $r(v)$'s are uniform, one obtains the $k$-Center
problem. Plesn\'ik [Plesn\'ik, Disc. Appl. Math. 1987] introduced the Priority
$k$-Center problem and gave a $2$-approximation algorithm matching the best
possible algorithm for $k$-Center. We show how the problem is related to two
different notions of fair clustering [Harris et al., NeurIPS 2018; Jung et al.,
FORC 2020]. Motivated by these developments we revisit the problem and, in our
main technical contribution, develop a framework that yields constant factor
approximation algorithms for Priority $k$-Center with outliers. Our framework
extends to generalizations of Priority $k$-Center to matroid and knapsack
constraints, and as a corollary, also yields algorithms with fairness
guarantees in the lottery model of Harris et al [Harris et al, JMLR 2019].
