Computations of Lipschitz summing norms and applications
Volume 165 / 2021
Abstract
We describe and analyze an algorithm to compute exactly Lipschitz -summing norms of maps between finite metric spaces. In contrast to the linear case, where even the computation of (p,\theta )-summing norms between finite-dimensional normed spaces is in general difficult, Lipschitz (p,\theta )-summing norms of maps between finite metric spaces can be reduced to the computation of extreme points of certain polyhedra and the subsequent solution of a finite linear program. The results of such computations when \theta =0 are used to provide counterexamples to a composition formula for Lipschitz p-summing maps, which solves the open problem stated by J. D. Farmer and W. B. Johnson in their seminal paper which introduced the notion of Lipschitz p-summing maps. We give some examples of computations of Lipschitz (p,\theta )-summing norms of graph metrics and present concluding remarks. Finally, we raise some open problems which we think are interesting.