Hilbert volume in metric spaces. I
Let \(Df\) denote the Jacobian matrix of a differentiable map \(f: \mathbb R^n\to\mathbb R^n\). One way to quantify the infinitesimal dilation of \(f\) is to consider the operator norm \(\|Df\|\). This corresponds to the local form of the Lipschitz constant \(\text{Lip}\,f=\sup_{a,b} \frac{|f(a)-f(b)|}{|a-b|}\), and thus makes sense in general metric spaces. Another natural, and often more convenient way to measure dilation is the Hilbert-Schmidt norm \(\|Df\|_{HS}\). However, the latter does not immediately generalize to metric spaces. The present article developes such a generalization and uses it to derive several previous results in a unified and elegant way. Instead of trying to describe the construction in full generality, let us consider the special case of Lipschitz maps \(f: X \to {\mathbb{R}}^n\), where \(X\) is a metric space. Let \(\mu\) be a measure on the set \(\mathcal P\) of all rank-\(1\) projections, which can be identified on the \((n-1)\)-dimensional projective space. Of particular importance are the measures \(\mu\) for which \(\int_{\mathcal P}p(x)\,d\mu = x\) for all \(x\in \mathbb R^n\). Such a measure is called an \textit{axial partition of unity}; a related term in harmonic analysis is a \textit{Parseval frame}. The \(L_2\)-dilation of \(f\) with respect to \(\mu\) is \[ \|\text{dil}^* f\|_{L_2(\mu)}=\left(\int_{\mathcal P} \text{Lip}^2(p\circ f) \,d\mu(p)\right)^{1/2}. \] In terms of frames, this definition means that one applies the analysis operator to \(f\) and measures the Lipschitz constant of the output. Another approach proceeds via the synthesis operator: One can consider all Lipschitz maps \(\tilde f: X\to L_2(\mathcal P,\mu)\) from which \(f\) can be synthesized and defines \[ \left\|\widetilde{\text{dil}}^* f\right\|_{L_2(\mu)} = \inf_{\tilde f} \left(\int_{\mathcal P} \text{Lip}^2(\tilde f(\cdot,p))\,d\mu(p) \right)^{1/2}. \] For every axial partition of unity one has \(\|\widetilde{\text{dil}}^* f\|_{L_2(\mu)} \leq \|\text{dil}^* f\|_{L_2(\mu)}\) since the composition of analysis and synthesis recovers \(f\). Taking the infimum over all axial partitions of unity \(\mu\) yields minimal \(L_2\)-dilations \(\|\min\text{dil}^* f\|_{L_2}\) and \(\|\min\widetilde{\text{dil}}^* f\|_{L_2}\). For linear maps between Euclidean spaces the minimal \(L_2\)-dilation of either kind is exactly the Hilbert-Schmidt norm. For non-linear maps they need to be localized first, by taking restrictions to small neighborhoods of a point. The concept turns out to be useful, e.g., for proving volume comparison theorems. The author proves an elegant form of F. John's ellipsoid theorem in terms of \(\|\min\text{dil}^* f\|_{L_2}\), recasts the Burago-Ivanov proof of the Hopf conjecture [\textit{D. Burago} and \textit{S. Ivanov}, Geom. Funct. Anal. 4, No. 3, 259--269 (1994; Zbl 0808.53038)] in these new terms, and presents further extensions and applications of his approach.
scientific article; zbMATH DE number 6039956
scientific article