  Friedman and Linial introduced the convex body chasing problem to explore the
interplay between geometry and competitive ratio in metrical task systems. In
convex body chasing, at each time step $t \in \mathbb{N}$, the online algorithm
receives a request in the form of a convex body $K_t \subseteq \mathbb{R}^d$
and must output a point $x_t \in K_t$. The goal is to minimize the total
movement between consecutive output points, where the distance is measured in
some given norm.
  This problem is still far from being understood, and recently Bansal et al.
gave an algorithm for the nested version, where each convex body is contained
within the previous one. We propose a different strategy which is $O(d \log
d)$-competitive algorithm for this nested convex body chasing problem,
improving substantially over previous work. Our algorithm works for any norm.
This result is almost tight, given an $\Omega(d)$ lower bound for the
$\ell_{\infty}$.
