On the coincidence of the canonical embeddings of a metric space into a Banach space
Let \((X,\rho)\) be a metric space and \(F(X)\) be the space of finitely supported signed measures on \(X\) with zero total charge (that is, taking zero values on constant functions). The author considers two norms on \(F(X)\). One of them, the Kantorovich-Rubinstein norm, is defined as \(\|\mu\|_{KR}=\sup |\int f\,d\mu|\), where the supremum is over all \(1\)-Lipschitz functions \(f:X\to\mathbb{R}\). The second, the author calls it the Hausdorff-Kuratowski norm, is defined as \(\|\mu\|_{HK}=\sup_{x\in X}|\int \rho(x,y)\,d\mu(y)|\). The main result of the paper says that the norms \(\|\mu\|_{KR}\) and \(\|\mu\|_{HK}\) are different for finite metric spaces \(X\) with at least \(5\) points. The main motivation for the paper is that the measures are related to two canonical ways of embedding a metric space into a Banach space. Reviewer's remarks: (1) The completion of the normed space \((F(X),\|\mu\|_{KR})\) is also known as the Arens-Eells space [see \textit{R.\,F.\thinspace Arens} and \textit{J.\,Eells}, Pac.\ J.\ Math.\ 6, 397--403 (1956; Zbl 0073.39601) and \textit{N.\,Weaver}, Lipschitz algebras.\ Singapore: World Scientific (1999; Zbl 0936.46002)]. Closely related objects are the Lipschitz-free space [see \textit{G.\,Godefroy} and \textit{N.\,J .\,Kalton}, Stud.\ Math.\ 159, No.\,1, 121--141 (2003; Zbl 1059.46058)] and the free Banach space [see \textit{V.\,G.\thinspace Pestov}, Funct.\ Anal.\ Appl.\ 20, 70--72 (1986); translation from Funkts.\ Anal.\ Prilozh.\ 20, No.\,1, 81--82 (1986; Zbl 0635.22005)]. (2) The embedding \(x\mapsto\rho(x,\cdot)\) into the space \(C(X)\), which the author calls the Hausdorff-Kuratowski embedding, is also called the Fréchet embedding, at least for finite metric spaces [see \textit{J.\,Matoušek}, ``Lectures on discrete geometry'' (Graduate Texts in Mathematics 212; New York/NY:\ Springer) (2002; Zbl 0999.52006)].
scientific article; zbMATH DE number 5632971