  Given a metric space, the $(k,z)$-clustering problem consists of finding $k$
centers such that the sum of the of distances raised to the power $z$ of every
point to its closest center is minimized. This encapsulates the famous
$k$-median ($z=1$) and $k$-means ($z=2$) clustering problems. Designing
small-space sketches of the data that approximately preserves the cost of the
solutions, also known as \emph{coresets}, has been an important research
direction over the last 15 years.
  In this paper, we present a new, simple coreset framework that simultaneously
improves upon the best known bounds for a large variety of settings, ranging
from Euclidean space, doubling metric, minor-free metric, and the general
metric cases.
