Lipschitz Selectors May Not Yield Competitive Algorithms for Convex Body Chasing

The current best algorithms for the convex body chasing (CBC) problem in online algorithms use the notion of the Steiner point of a convex set. In particular, the algorithm that always moves to the Steiner point of the request set is O(d) competitive for nested CBC, and this is optimal among memoryless algorithms [Bubeck et al.: Chasing nested convex bodies nearly optimally. In: 31st Annual ACM–SIAM Symposium on Discrete Algorithms (Salt Lake City 2020), pp. 1496–1508. SIAM, Philadelphia (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 CBC. We answer this question in the negative by exhibiting a selector that yields a non-competitive algorithm for nested CBC but is Lipschitz with respect to Hausdorff distance. Furthermore, we show that being Lipschitz with respect to an L_p -type analog of the Hausdorff distance is sufficient to guarantee competitiveness if and only if p= 1 .