  $\renewcommand{\Re}{\mathbb{R}} \newcommand{\eps}{\varepsilon}
  \newcommand{\Net}{S}
  \newcommand{\tldO}{{\widetilde{O}}}
  \newcommand{\body}{C}
  $ We revisit an algorithm of Clarkson etal [CEMST96], that computes (roughly)
a $1/(4d^2)$-centerpoint in $\tldO(d^9)$ time, for a point set in $\Re^d$,
where $\tldO$ hides polylogarithmic terms. We present an improved algorithm
that computes (roughly) a $1/d^2$-centerpoint with running time $\tldO(d^7)$.
While the improvements are (arguably) mild, it is the first progress on this
well known problem in over twenty years. The new algorithm is simpler, and the
running time bound follows by a simple random walk argument, which we believe
to be of independent interest. We also present several new applications of the
improved centerpoint algorithm.
