Power-Aggregation of Pseudometrics and the McShane-Whitney Extension Theorem for Lipschitz p $p$ -Concave Maps
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Power-Aggregation of Pseudometrics and the McShane-Whitney Extension Theorem for Lipschitz p-Concave Maps J. Rodríguez-López1 · E.A. Sánchez-Pérez1
Received: 13 March 2020 / Accepted: 4 July 2020 © Springer Nature B.V. 2020
Abstract Given a countable set of families {Dk : k ∈ N} of pseudometrics over the same set D, we study the power-aggregations of this class, that are defined as convex combinations of integral averages of powers of the elements of ∪k Dk . We prove that a Lipschitz function f is dominated by such a power-aggregation if and only if a certain property of super-additivity involving the powers of the elements of ∪k Dk is fulfilled by f . In particular, we show that a pseudo-metric is equivalent to a power-aggregation of other pseudometrics if this kind of domination holds. When the super-additivity property involves a p-power domination, we say that the elements of Dk are p-concave. As an application of our results, we prove under this requirement a new extension result of McShane-Whitney type for Lipschitz p-concave real valued maps. Keywords Aggregation · Pseudometric · Lipschitz function · Extension · p-average Mathematics Subject Classification Primary 46N10 · Secondary 54A10 · 26A16 · 54E45
1 Introduction One of the main tools for multi-objective optimization is the aggregation of real valued (objective) functions for obtaining a new real valued objective map. The aggregation function usually represents the way the decision maker wants to combine the different objectives of the multi-valued optimization problem for getting a meaningful solution. In the basic cases, Both authors gratefully acknowledge the support of the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigaciones and FEDER under each grants MTM2015-64373-P (MINECO/FEDER, UE) and MTM2016-77054-C2-1-P.
B E.A. Sánchez-Pérez
[email protected] J. Rodríguez-López [email protected]
1
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
J. Rodríguez-López, E.A. Sánchez-Pérez
these functions are weighted linear combinations as well as p-sums of the set of original functions. In case the objective functions are metrics, we have a classical optimization problem. Particular cases in which the aggregation function is given by a weighted p-norm —if the elements of the metric space constitute a subset of a linear space—, have already become classical tools in multi-objective optimization (see for example [2, 14], and the notion of Lipschitz p-stability in [3, 1.3.5.]). It is also on the basis of more sophisticated methods, as the so-called OWA (Ordered Weighted Averaging) (see [3, 1.4.4]). Motivated in part by this generalized use of averages and p-norms, we are interested in showing a link among this applied context and some classical tools coming from pure topology and functional analysis, that would contribute with new ideas for the foundations of the multi-objective optimization and to widen the set of theoretical tools in
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