  We study the problem of chasing convex bodies online: given a sequence of
convex bodies $K_t\subseteq \mathbb{R}^d$ the algorithm must respond with
points $x_t\in K_t$ in an online fashion (i.e., $x_t$ is chosen before
$K_{t+1}$ is revealed). The objective is to minimize the sum of distances
between successive points in this sequence. Bubeck et al. (STOC 2019) gave a
$2^{O(d)}$-competitive algorithm for this problem. We give an algorithm that is
$O(\min(d, \sqrt{d \log T}))$-competitive for any sequence of length $T$.
