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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.
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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.
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.
Let $D$ denote the distance matrix for an $n+1$ point metric space $(X,d)$. In the case that $X$ is an unweighted metric tree, the sum of the entries in $D^{-1}$ is always equal to $2/n$. Such trees can be considered as affinely independent subsets of the Hamming cube $H_n$, and it was conjectured that the value $2/n$ was minimal among all such subsets. In this paper we confirm this conjecture and give a geometric interpretation of our result which applies to any subset of $H_n$.
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.
We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. We also point out a counterexample, due to F. Nazarov, to a previous conjecture that that triangulated packings with fixed numbers of disks with fixed numbers of disks for each radius claiming that such packings were the most dense.
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