Title: Nested Convex Bodies are Chaseable,
Abstract: In the Convex Body Chasing problem, we are given an initial point $v_0$ in
$R^d$ and an online sequence of $n$ convex bodies $F_1, ..., F_n$. When we
receive $F_i$, we are required to move inside $F_i$. Our goal is to minimize
the total distance travelled. This fundamental online problem was first studied
by Friedman and Linial (DCG 1993). They proved an $\Omega(\sqrt{d})$ lower
bound on the competitive ratio, and conjectured that a competitive ratio
depending only on d is possible. However, despite much interest in the problem,
the conjecture remains wide open.
We consider the setting in which the convex bodies are nested: $F_1 \supset
... \supset F_n$. The nested setting is closely related to extending the online
LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints.
Moreover, this setting retains much of the difficulty of the general setting
and captures an essential obstacle in resolving Friedman and Linial's
conjecture. In this work, we give the first $f(d)$-competitive algorithm for
chasing nested convex bodies in $R^d$.