Journey to the Center of the Point Set

$\renewcommand{\Re}{$\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 $1\mathrm{/}(4d^2)$-centerpoint in $\text{tldO}(d^9)$ time, for a point set in ${\mathrm{\Re }}^d$, where $\text{tldO}$ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a $1\mathrm{/}{d}^2$-centerpoint with running time $\text{tldO}(d^7)$. 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.