# 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.