Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces

We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in Fractured fractals and broken dreams by David and Semmes, or equivalently, Question 22 and hence also Question 24 in Thirty-three yes or no questions about mappings, measures, and metrics by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors 4-regular metric space X having big pieces of itself such that no Lipschitz map from a subset of X to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from the Heisenberg group to the space X having as image the whole X. As part of proving the above result we define a new distance on the Heisenberg group that is bounded by the Carnot-Carathéodory distance, which preserves the Ahlfors-regularity, and such that the Carnot-Carathéodory distance and the new distance are biLipschitz equivalent on no set of positive measure. This construction works more generally in any Ahlfors-regular metric space where one can make suitable shortcuts. Such spaces include, for example, all snowflaked Ahlfors-regular metric spaces. With the same techniques we also provide an example of a left-invariant distance on the Heisenberg group biLipschitz to the Carnot-Carathéodory distance for which no blow-up admits non-trivial dilations

Impossibility of dimension reduction in the nuclear norm

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$. We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $\mathcal{C}\subseteq \mathsf{S}_1$ with $|\mathcal{C}|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. $\mathcal{C}$ is not even a $O(1)$-Lipschitz quotient of any subset of any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. Thus, $\mathsf{S}_1$ does not admit a dimension reduction result \'a la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with $\mathsf{S}_1$ replaced by the Banach space $\ell_1$ of absolutely summable sequences via the work of Brinkman and Charikar (2003). In fact, the above set $\mathcal{C}$ can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in $\mathsf{S}_1$. The challenge is to demonstrate that $\mathcal{C}$ cannot be faithfully realized in an arbitrary low-dimensional subspace of $\mathsf{S}_1$, while Brinkman and Charikar obtained such an assertion only for subspaces of $\mathsf{S}_1$ that consist of diagonal operators (i.e., subspaces of $\ell_1$). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace $X$ of $\mathsf{S}_1$ is at most a universal constant multiple of $\sqrt{\log \mathrm{dim}(X)}$