A Menger Redux: Embedding Metric Spaces Isometrically in Euclidean Space

We present geometric proofs of Menger’s results on isometrically embedding metric spaces in Euclidean space. In 1928, Karl Menger [6] published the proof of a beautiful characterization of those metric spaces that are isometrically embeddable in the ndimensional Euclidean space E. While a visitor at Harvard University and the Rice Institute in Houston during the 1930-31 academic year, Menger gave courses on metric geometry in which he “considerably shortened and revised [his] original proofs and generalized the formulation.” The new proofs of the 1930-31 academic year appear in the English language article New foundation of Euclidean geometry [7] published in the American Journal of Mathematics in 1931. Leo Liberti and Carlile Lavor in their beautifully written article Six mathematical gems from the history of distance geometry remark on their review of a part of Menger’s characterization that “it is remarkable that almost none of the results below offers an intuitive geometrical grasp, such as the proofs of Heron’s formula and the Cayley Theorem do. As formal mathematics has it, part of the beauty in Menger’s work consists in turning the ‘visual’ geometrical proofs based on intuition into formal symbolic arguments based on sets and relations.” Part of the reason that Menger’s results fail to offer an “intuitive geometric grasp” is that Menger offers his results in the general setting of an abstract congruence system and semi-metric spaces. We believe that Menger’s characterization deserves a wider circulation among the mathematics community. The aim of this article is to explicate Menger’s characterization in the category of metric spaces rather than in his original setting of congruence systems and their model semi-metric spaces, and in doing so to give a mostly self-contained treatment with straightforward geometric proofs of his characterization. We provide all the background material on the geometry of Euclidean space that is needed to prove Menger’s results so that the proofs are accessible to any undergraduate who has mastered the basics of real linear algebra and real inner product 1[7], p. 721. 2[4], p. 12. 3See footnote 5.