  The current best algorithms for convex body chasing problem in online
algorithms use the notion of the Steiner point of a convex set. In particular,
the algorithm which always moves to the Steiner point of the request set is
$O(d)$ competitive for nested convex body chasing, and this is optimal among
memoryless algorithms [Bubeck et al. 2020]. A memoryless algorithm coincides
with the notion of a selector in functional analysis. The Steiner point is
noted for being Lipschitz with respect to the Hausdorff metric, and for
achieving the minimal Lipschitz constant possible. It is natural to ask whether
every selector with this Lipschitz property yields a competitive algorithm for
nested convex body chasing. We answer this question in the negative by
exhibiting a selector which yields a non-competitive algorithm for nested
convex body chasing but is Lipschitz with respect to Hausdorff distance.
Furthermore, we show that being Lipschitz with respect to an $L_p$-type analog
to the Hausdorff distance is sufficient to guarantee competitiveness if and
only if $p=1$.
