A Linear Lower Bound for Convex Body Chasing in Every $\ell_p$ Norm

We establish an $\Omega(d)$ lower bound on the competitive ratio of convex body chasing in $\mathbb{R}^d$ with $\ell_p$ movement costs for any $p \in [1, \infty)$. Previously, the best known lower bound for general $p$ was $\Omega(\max\{\sqrt{d}, d^{1-1/p}\})$, established by Bubeck et al. (2019); a linear lower bound was known only for $p = \infty$, realized by the standard construction of chasing faces of a hypercube. Our lower bound strictly improves on the prior bound for every fixed $p \in [1, \infty)$ and shows that the functional Steiner point algorithm of Sellke (2023) - which has competitive ratio $O(d)$ - is asymptotically optimal for every $\ell_p$ norm. Our lower bound is based on the recursive construction of Bubeck, Coester, and Rabani (2023) for metrical service systems, which gives a lower bound of $\Omega(r^2)$ for requests of size $O(r)$ in that setting. We show that requests in their construction, each comprising a set of discrete points in a graph metric, can be mapped to the vertices of convex requests in a $(2r+1)$-dimensional vector space while approximately preserving movement costs up to a factor of $O(r)$, which establishes the lower bound.