Lipschitz Selectors May Not Yield Competitive Algorithms for Convex Body Chasing

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 [BKL+20]. 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
Lp-type analog to the Hausdorff distance is sufficient to guarantee competitiveness if and only
if p = 1.