  Given a set of points of interest, a volumetric spanner is a subset of the
points using which all the points can be expressed using "small" coefficients
(measured in an appropriate norm). Formally, given a set of vectors $X = \{v_1,
v_2, \dots, v_n\}$, the goal is to find $T \subseteq [n]$ such that every $v
\in X$ can be expressed as $\sum_{i\in T} \alpha_i v_i$, with $\|\alpha\|$
being small. This notion, which has also been referred to as a well-conditioned
basis, has found several applications, including bandit linear optimization,
determinant maximization, and matrix low rank approximation. In this paper, we
give almost optimal bounds on the size of volumetric spanners for all $\ell_p$
norms, and show that they can be constructed using a simple local search
procedure. We then show the applications of our result to other tasks and in
particular the problem of finding coresets for the Minimum Volume Enclosing
Ellipsoid (MVEE) problem.
