# Chasing Convex Bodies Optimally

## TL;DR

This paper introduces an improved online algorithm for the convex body chasing problem, achieving optimal or near-optimal competitive ratios in various normed spaces, advancing the understanding of online convex optimization.

## Contribution

The authors develop a new algorithm with a competitive ratio of d in any normed space, and O(√(d log N)) in Euclidean space, improving previous bounds and extending prior work on nested convex bodies.

## Key findings

- Achieves competitive ratio d in any normed space, tight for ℓ∞.
- Attains competitive ratio O(√(d log N)) in Euclidean space.
- Extends prior work using the functional Steiner point of convex functions.

## Abstract

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. We give an improved algorithm achieving competitive ratio $d$ in any normed space, which is exactly tight for $\ell^{\infty}$. In Euclidean space, our algorithm also achieves competitive ratio $O(\sqrt{d\log N})$, nearly matching a $\sqrt{d}$ lower bound when $N$ is subexponential in $d$. The approach extends our prior work for nested convex bodies, which is based on the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the associated work function.