Computations of Lipschitz summing norms and applications
We describe and analyze an algorithm to compute exactly Lipschitz $(p,\theta )$-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.