Power-Aggregation of Pseudometrics and the McShane-Whitney Extension Theorem for Lipschitz $p$-Concave Maps

Abstract

Given a countable set of families $\{\mathcal{D}_{k}:k \in \mathbb{N}\}$ 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 $\cup _{k} \mathcal{D}_{k}$. 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 $\cup _{k} \mathcal{D}_{k}$ 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 $\mathcal{D}_{k}$ 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.