Mazur Distance Research Articles

Estimation of the centroid Banach–Mazur distance between planar convex bodies

UDC 514.18 We consider a version of the Banach–Mazur distance between two convex bodies C and D from E d with an additional requirement that their centroids coincide. We prove that for any two convex bodies C and D in E 2 .

On the Banach–Mazur Distance in Small Dimensions

AbstractWe establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in $$\mathbb {R}^3$$ R 3 and $$\mathbb {R}^4$$ R 4 . In dimension three this distance is equal to $$\frac{9}{5}$$ 9 5 , and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.

The Banach–Mazur distance between isomorphic spaces of continuous functions is not always an integer

For over half a century, Aleksander Pełczyński's question, whether the Banach–Mazur distance between any two isomorphic C(K) spaces is an integer, has remained open; as usual, K is a compact Hausdorff space and C(K) denotes the Banach space of all continuous real-valued functions on K, provided with the maximum norm. We answer this question in the negative. Moreover, we prove that the Banach–Mazur distance between isomorphic spaces C(K) and C0(L) is not always an integer; here L is a locally compact but noncompact Hausdorff space and C0(L) denotes the Banach space of all continuous real-valued functions vanishing at infinity on L, furnished with the maximum norm.

Banach–Mazur Distance from $$\ell_p^3$$ to $$\ell_\infty^3$$

Banach–Mazur Distance from $$\ell_p^3$$ to $$\ell_\infty^3$$

Lipschitz-free spaces and subsets of finite-dimensional spaces

We consider two questions on the geometry of Lipschitz-free $p$ -spaces $\mathcal {F}_p$ , where $0< p\leq 1$ , over subsets of finite-dimensional vector spaces. We solve an open problem and show that if $(\mathcal {M}, \rho )$ is an infinite doubling metric space (e.g. an infinite subset of an Euclidean space), then $\mathcal {F}_p (\mathcal {M}, \rho ^\alpha )\simeq \ell _p$ for every $\alpha \in (0,\,1)$ and $0< p\leq 1$ . An upper bound on the Banach–Mazur distance between the spaces $\mathcal {F}_p ([0, 1]^d, |\cdot |^\alpha )$ and $\ell _p$ is given. Moreover, we tackle a question due to Albiac et al. [4] and expound the role of $p$ , $d$ for the Lipschitz constant of a canonical, locally coordinatewise affine retraction from $(K, |\cdot |_1)$ , where $K=\cup _{Q\in \mathcal {R}} Q$ is a union of a collection $\emptyset \neq \mathcal {R} \subseteq \{ Rw + R[0,\,1]^d: w\in \mathbb {Z}^d\}$ of cubes in $\mathbb {R}^d$ with side length $R>0$ , into the Lipschitz-free $p$ -space $\mathcal {F}_p (V, |\cdot |_1)$ over their vertices.

Isomorphisms of {mathcal {C}}(K, E) Spaces and Height of K

Let K_1, K_2 be compact Hausdorff spaces and E_1, E_2 be Banach spaces not containing a copy of c_0. We establish lower estimates of the Banach–Mazur distance between the spaces of continuous functions {mathcal {C}}(K_1, E_1) and {mathcal {C}}(K_2, E_2) based on the ordinals ht(K_1), ht(K_2), which are new even for the case of spaces of real-valued functions on ordinal intervals. As a corollary we deduce that {mathcal {C}}(K_1, E_1) and {mathcal {C}}(K_2, E_2) are not isomorphic if ht(K_1) is substantially different from ht(K_2).

A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

We prove that if \(X\) is an \(\ell_{1}\)-predual isomorphic to the space \(c_{0}\) of sequences converging to zero, then for any isomorphism \(T:X\rightarrow c_{0}\) we have \(\Vert T\Vert\, \Vert T^{-1}\Vert\ge1+2r^{*}(X)\), where \(r^{*}(X)\) is the smallest radius of the closed ball of the dual space \(X^{*}\) containing all the weak\(^{*}\) cluster points of the set of all extreme points of the closed unit ball of \(X^*\).

Volume ratios for Cartesian products of convex bodies

The paper is devoted to the behavior of volume ratios, the modified Banach–Mazur distance, and the vertex index for sums of convex bodies. It is shown that sup d ( A ⊕ K , B ⊕ L ) ≥ sup ∂ ( A ⊕ K , B ⊕ L ) ≥ c ⋅ n 1 − k + k ′ 2 n , \begin{equation*} \sup d (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq \sup \partial (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq c \cdot n^{1-\frac {k+k’}{2n}}, \end{equation*} if K ⊂ R n \mathrm {K}\subset \mathbb {R}^n and L ⊂ R k \mathrm {L}\subset \mathbb {R}^k are convex and symmetric (the supremum is taken over all symmetric convex bodies A ⊂ R n − k \mathrm {A}\subset \mathbb {R}^{n-k} and B ⊂ R n − k ′ ) \mathrm {B}\subset \mathbb {R}^{n-k’}) . Furthermore, some examples are discussed that show that the available extimates of the vertex index in terms of the volume ratio are not sharp.

Relative growth rate and contact Banach–Mazur distance

In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by \(d_\mathrm{CBM}\). Explicitly, we consider the following two set-ups, either on a contact manifold \(W \times S^1\) where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of \(d_\mathrm{CBM}\) are different in these two cases. In the former case the inputs are (contact) star-shaped domains of \(W \times S^1\) which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance \(d_\mathrm{CBM}\) defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of \(d_\mathrm{CBM}\). In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira’s sheaf quantization.

Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text]. The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text]. Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of [Formula: see text] into this space for all [Formula: see text]. On the other hand, in the case of domains in [Formula: see text], we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

Banach\u2013Mazur Distance from the Parallelogram to the Affine-Regular Hexagon and Other Affine-Regular Even-Gons

We show that the Banach–Mazur distance between the parallelogram and the affine-regular hexagon is frac{3}{2} and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just frac{3}{2}. A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach–Mazur distance of any planar centrally-symmetric bodies is at most frac{3}{2}. Analogously, we deal with the Banach–Mazur distances between the parallelogram and the remaining affine-regular even-gons.

Banach–Mazur stability of von Neumann algebras

We initiate the study of perturbation of von Neumann algebras relatively to the Banach–Mazur distance. We first prove that the type decomposition is continuous, i.e. if two von Neumann algebras are close, then their respective summands of each type are close. We then prove that, under some vanishing conditions on its Hochschild cohomology groups, a von Neumann algebra is Banach–Mazur stable, i.e. any von Neumann algebra which is close enough is actually Jordan ∗-isomorphic. These vanishing conditions are possibly empty.

Approximation of the average of some random matrices

Rudelson's theorem states that if for a set of unit vectors ui and positive weights ci, we have that ∑ciui⊗ui is the identity operator I on Rd, then the sum of a random sample of Cdln⁡d of these diadic products is close to I. The ln⁡d term cannot be removed.On the other hand, the recent fundamental result of Batson, Spielman and Srivastava and its improvement by Marcus, Spielman and Srivastava show that the ln⁡d term can be removed, if one wants to show the existence of a good approximation of I as the average of a few diadic products. It is known that essentially the same proof as Rudelson's yields a more general statement about the average of positive semi-definite matrices.First, we give an example of an average of positive semi-definite matrices where there is no approximation of this average by Cd elements. Thus, the result of Batson, Spielman and Srivastava cannot be extended to this wider class of matrices.Next, we present a stability version of Rudelson's result on positive semi-definite matrices, and thus, extend it to certain non-symmetric matrices. This yields applications to the study of the Banach–Mazur distance of convex bodies.Finally, we show that in some cases, one needs to take a subset of the vectors of order d2 to approximate the identity.

Symplectic Banach–Mazur distances between subsets of ℂn

Following proposals of Ostrover and Polterovich, we introduce and study “coarse” and “fine” versions of a symplectic Banach–Mazur distance on certain open subsets of [Formula: see text] and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljević and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach–Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in [Formula: see text] admits quasi-isometric embeddings of [Formula: see text] for every finite dimension D. Our constructions are obtained from a general method of constructing [Formula: see text]-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given [Formula: see text]-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from [J. Gutt and M. Usher, Symplectically knotted codimension-zero embeddings between domains in [Formula: see text], Duke Math. J. 168 (2019) 2299–2363].

Greedy algorithms and Kolmogorov widths in Banach spaces

Let X be a Banach space and K be a compact subset in X. We consider a greedy algorithm for finding n-dimensional subspace Vn⊂X which can be used to approximate the elements of K. We are interested in how well the space Vn approximates the elements of K. For this purpose we compare the performance of greedy algorithm measured by σn(K)X≔dist(K,Vn)X with the Kolmogorov width dn(K)X which is the best possible error one can achieve when approximating K by n-dimensional subspaces. Various results in this direction have been given, e.g., in Binev et al. (2011), DeVore et al. (2013) and Wojtaszczyk (2015). The purpose of the present paper is to continue this line. We shall show that there exists a constant C>0 such that σn(K)X≤Cn−s+μ(log(n+1))min(s,1∕2),n≥1, if Kolmogorov widths dn(K)X decay as n−s and the Banach–Mazur distance between an arbitrary n-dimensional subspace Vn⊂X and ℓ2n satisfies d(Vn,ℓ2n)≤C1nμ. In particular, when some additional information about the set K is given then there is no logarithmic factor in this estimate.

On the Banach–Mazur distance to cross-polytope

Let n≥3, and let B1n be the standard n-dimensional cross-polytope (i.e. the convex hull of standard coordinate vectors and their negatives). We show that there exists a symmetric convex body Gm in Rn such that the Banach–Mazur distance dBM(B1n,Gm) satisfies dBM(B1n,Gm)≥n5/9log−C⁡n, where C>0 is a universal constant. The body Gm is obtained as a typical realization of a random polytope in Rn with 2m:=2n3 vertices. The result improves upon an earlier estimate of S. Szarek which gives dBM(B1n,Gm)≥cn1/2log⁡n (with a different choice of m). This shows in a strong sense that the cross-polytope (or the cube [−1,1]n) cannot be an “approximate” center of the Minkowski compactum.

A norm inequality in James’ space and stability of the fixed point property

In this paper we prove a norm inequality in James’ space J, and use it to show that the fixed point property for nonexpansive mappings is passed on from J to those Banach spaces X whose Banach–Mazur distance to J satisfies $$d(X,J)<\sqrt{\frac{17+\sqrt{97}}{12}}$$ .

On Banach space properties that are not invariant under the Banach–Mazur distance 1

We consider which properties of Banach spaces are not invariant under the Banach–Mazur distance 1; we will say that a given Property P is not invariant under the Banach–Mazur distance 1 if there exist two Banach spaces X and Y such that X has Property P, Y fails Property P, and the Banach–Mazur distance d(X,Y)=1. The main results of this paper concern the case of separable Lindenstrauss spaces. First, in the setting of preduals of ℓ1, we give some geometric equivalences for polyhedral properties. Next, we show that, even in the restricted framework of preduals of ℓ1, most of the notions of polyhedral Banach spaces (and their geometric equivalences) fail to be invariant under the Banach–Mazur distance 1. Then, in the general case of Banach spaces, we indicate some other geometrical properties that are invariant or not invariant under the Banach–Mazur distance 1.

On Banach–Mazur distance between planar convex bodies

Upper estimates of the diameter and the radius of the family of all planar convex bodies with respect to the Banach-Mazur distance are obtained. Namely, it is shown that the diameter does not exceed $\tfrac{19-\sqrt{73}}4\approx 2.614$, which improves the previously known bound of $3$, and that the radius does not exceed $\frac{117}{70}\approx 1.671$.

The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, e.g. concerning the Helly dimension or the Banach–Mazur distance.