Almost isometries and orthogonality (Q556236)

Let \(A\) be an orthonormal basis of \(\mathbb R^n\). It follows from the flattening lemma of Johnson and Lindenstrauss [\textit{W. B. Johnson} and \textit{J. Lindenstrauss}, Contemp. Math. 26, 189--206 (1984; Zbl 0539.46017)] and the concentration of measure on the sphere that there is an \(m > 0\) and a \((1 + \varepsilon)\)-bi-Lipschitz mapping \(T : A \rightarrow \mathbb R^d\) where \(d \leq c \varepsilon^{-2} \log n\) and \(T = m P\) (\(P\) is the orthogonal projection onto \(\mathbb R^d\) and \(c>0\) is an absolute constant). This paper positively answers the natural and important question if \(T^{-1}\) can be extended to a \((1 + \delta(\varepsilon))\)-bi-Lipschitz map on \(\mathbb R^n\). Let \(X\) be a Hilbert space with unit sphere \(S\). \(A \subset S\) is said to be \(\varepsilon\)-almost orthonormal for some \(0< \varepsilon <1\) if \(| \langle x,y\rangle| \leq \varepsilon\) for all \(x \neq y \in A\). Following the suggestion of the author (``a reader interested just in the basic extension method''), we will only report on this aspect of the main theorem (Theorem 5.1) in this short review. There exists \(\varepsilon_1 > 0\) and a continuous increasing function \(\delta: [0,\varepsilon_1] \rightarrow [0,1]\) such that \(\delta(0) = 0\) with the following property. Let \(X\), \(S\) be as above. Suppose \(A, A^{\sim} \subset S\) such that \(A \cup A^{\sim}\) is \(\varepsilon\)-almost orthonormal. Let \(F: A \rightarrow A^{\sim}\) be a bijection. Then \(F\) admits a \((1+\delta(\varepsilon))\)-bilipschitz extension \(F: X \rightarrow X\) homotopic to the identity.