Chasing Convex Bodies Optimally
In the chasing convex bodies problem, an online player receives a request sequence of $N$ convex sets $K_1,\dots, K_N$ contained in a normed space $\mathbb R^d$. The player starts at $x_0\in \mathbb R^d$, and after observing each $K_n$ picks a new point $x_n\in K_n$. At each step the player pays a movement cost of $||x_n-x_{n-1}||$. The player aims to maintain a constant competitive ratio against the minimum cost possible in hindsight, i.e. knowing all requests in advance. The existence of a finite competitive ratio for convex body chasing was first conjectured in 1991 by Friedman and Linial. This conjecture was recently resolved with an exponential $2^{O(d)}$ upper bound on the competitive ratio. 
In this paper, we drastically improve the exponential upper bound. We give an algorithm achieving competitive ratio $d$ for arbitrary normed spaces, which is exactly tight for $\ell^{\infty}$. In Euclidean space, our algorithm achieves nearly optimal competitive ratio $O(\sqrt{d\log N})$, compared to a lower bound of $\sqrt{d}$. Our approach extends another recent work which chases nested convex bodies using the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the work function to obtain our algorithm.