Union of Euclidean metric spaces is Euclidean (Q2826222)

The authors introduce a number of concepts. ``We denote the \(k\)-dimensional Euclidean space by \(\ell_2^k\) and the (separable) Hilbert space by \(\ell_2^\infty\). The Lipschitz constant of a map \(f\) from a metric space \((X,d_X)\) to a metric space \((Y,d_Y)\) is \(\| f\|_{\mathrm{Lip}}=\sup_{x,y\in X}\frac{d_Y(f(x),f(y))}{d_X(x,y)}\). The distortion of an embedding \(f:X\hookrightarrow Y\) is \(\| f\|_{\mathrm{Lip}}\| f^{-1}\|_{\mathrm{Lip}}\) (where \(f^{-1}\) is the inverse map from \(f(X)\subset Y\) to \(X\)).''NEWLINENEWLINEThe main result that the authors obtain is the following theorem: ``Consider a metric space \((X,d)\). Assume that \(X\) is the union of two metric subspaces \(A\) and \(B\) that embed into \(\ell_2^a\) and \(\ell_2^b\) with distortions \(D_A\) and \(D_B\), respectively. Then \(X\) embeds into \(\ell_2^{a+b+1}\) with distortion at most \(7D_AD_B+2(D_A+D_B)\). If \(D_A=D_B=1\), then \(X\) embeds into \(\ell_2^{a+b+1}\) with distortion at most \(8.93\). In this theorem, \(a\) and \(b\) may be finite or infinite.''NEWLINENEWLINEThe authors deal also with topics like lower bounds on distortio, bi-Lipschitz extensions, an analog of the above theorem for arbitrary normed spaces, open problems.