Gromov’s “Hilbert volume in metric spaces I”

Just finished writing a summary of Gromov’s Hilbert Volume in Metric Spaces, Part 1 for Zentralblatt. Not an easy task to summarize such an article, and I essentially limited myself to the introductory part. But at least I contributed a Parseval frame analogy, which is not explicit in the article. I like frames in general, and tight/Parseval frames most of all.

And since the Zentralblatt server for review submission is down at the moment, the outlet for this text defaults to my blog.

Let Df denote the Jacobian matrix of a differentiable map f\colon \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 \mathrm{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 known previously 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\colon X\mapsto \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 axial partition of unity; a related term in harmonic analysis is a Parseval frame. The L_2-dilation of f with respect to \mu is \|\mathrm{dil}^* f\|_{L_2(\mu)}=\left(\int_{\mathcal P} \mathrm{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 is to use the synthesis operator: consider all Lipschitz maps \tilde f\colon X\to L_2(\mathcal P,\mu) from which f can be synthesized and define \|\widetilde{\mathrm{dil}}^* f\|_{L_2(\mu)}=\inf_{\tilde f} \left(\int_{\mathcal P} \mathrm{Lip}^2(\tilde f(\cdot,p))\,d\mu(p) \right)^{1/2}.

For every axial partition of unity one has \|\widetilde{\mathrm{dil}}^* f\|_{L_2(\mu)} \le \|\mathrm{dil}^* f\|_{L_2(\mu)} because the composition of analysis and synthesis recovers f. Taking the infimum over all axial partitions of unity \mu yields minimal L_2-dilations \|\min\mathrm{dil}^* f\|_{L_2} and \|\min\widetilde{\mathrm{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\mathrm{dil}^* f\|_{L_2}, recasts the Burago-Ivanov proof of the Hopf conjecture [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.