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Computations of Lipschitz summing norms and applications

Volume 165 / 2021

Manaf Adnan Saleh Saleh Colloquium Mathematicum 165 (2021), 31-40 MSC: Primary 47L20, 54E40; Secondary 26A16, 90-XX. DOI: 10.4064/cm8096-3-2020 Published online: 9 November 2020


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.


  • Manaf Adnan Saleh SalehDepartment of Mathematics and Computer Applications College of Science
    Al-Nahrain University
    Baghdad, Iraq

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