Chasing Convex Bodies with Linear Competitive Ratio

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 $T$.