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Register a new userIsometric embeddings of snowflakes into finite-dimensional Banach spaces
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We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.
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Let $V$ be a Banach space where for fixed $n$, $1<n<dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for even $n$ and all $V$. In this paper we give a positive answer for real $V$ and odd $n$ of the form $n=4k+1$, with the possible exception of $n=133.$ Our proof relies on a new characterization of ellipsoids in ${mathbb{R}}^n$, $ngeq 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.
We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of $SL(n,mathbb R)$ into $Sp(2(n-1),mathbb R)$ when no isometric embeddings exist. A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically this lemma is replaced by the quasi-flat theorem which says that maximal quasi-flat is within bounded distance of a finite union of flats. We improve this by showing that the quasi-flat is in fact flat off of a subset of codimension $2$.
Within the class of reflexive Banach spaces, we prove a metric characterization of the class of asymptotic-$c_0$ spaces in terms of a bi-Lipschitz invariant which involves metrics that generalize the Hamming metric on $k$-subsets of $mathbb{N}$. We apply this characterization to show that the class of separable, reflexive, and asymptotic-$c_0$ Banach spaces is non-Borel co-analytic. Finally, we introduce a relaxation of the asymptotic-$c_0$ property, called the asymptotic-subsequential-$c_0$ property, which is a partial obstruction to the equi-coarse embeddability of the sequence of Hamming graphs. We present examples of spaces that are asymptotic-subsequential-$c_0$. In particular $T^*(T^*)$ is asymptotic-subsequential-$c_0$ where $T^*$ is Tsirelsons original space.
We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $mathbb{F}^k$, with $mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k geqslant 2$ the metric subspace ${ | cdot |_p : p in [1,infty] }$ maps isometrically and monotonically to $[0, log k]$ (or $[0,1]$ by scaling the norm), again unlike in the Banach-Mazur compactum. Our analysis goes through embedding the above quotient space into a normed space, and reveals an implicit functorial construction of function spaces with diameter norms (as well as a variant of the distortion). In particular, we realize the above quotient space of norms as a normed space. We next study the parallel setting of the - also hitherto unexplored - metric space $mathcal{S}([n])$ of all metrics on a finite set of $n$ elements, revealing the connection between log-distortion and diameter norms. In particular, we show that $mathcal{S}([n])$ is also a normed space. We demonstrate embeddings of equivalence classes of finite metric spaces (parallel to the Gromov-Hausdorff setting), as well as of $mathcal{S}([n-1])$, into $mathcal{S}([n])$. We conclude by discussing extensions to norms on an arbitrary Banach space and to discrete metrics on any set, as well as some questions in both settings above.
Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2, that $Dsubset E$ and $Dsubset E$ are domains, and that $f: Dto D$ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform domains: Suppose that $f$ is a freely quasiconformal mapping and that $D$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Vaisala in the affirmative.
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