Banach-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.

The Banach-Mazur distance between C(Δ) and C_0(Δ) equals 2

Let $C(\Delta)$ denote the Banach space of all continuous real-valued functions on the Cantor set $\Delta$ and $C_0(\Delta)=\lbrace f\in C(\Delta): f(1)=0\rbrace$. From the 1966 theorem of Cambern, it is well-known that the Banach-Mazur distance $d(C(\Delta), C_0(\Delta))\geq 2$. We prove that, in fact, $d(C(\Delta), C_0(\Delta))= 2$. As a consequence, we answer a question left open in the 2012 paper of Candido and Galego.

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

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

On a colorful problem by Dol'nikov concerning translates of convex bodies

In this note we study a conjecture by Jerónimo-Castro, Magazinov and Soberón which generalized a question posed by Dol'nikov. Let F1,F2,…,Fn be families of translates of a convex compact set K in the plane so that each two sets from distinct families intersect. We show that, for some j, ⋃i≠jFi can be pierced by at most 4 points. To do so, we use previous ideas from Gomez-Navarro and Roldán-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square.

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.

A remark on geodesics in the Banach-Mazur distance

We show that there are uncountably many geodesics between any two non-isometric n n -dimensional normed spaces. We construct two explicit geodesics that can be used to describe all the points of the other geodesics.

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).

On the quasi-Figiel problem and extension of ε-isometry on unit sphere of [formula omitted] space

This paper aims to study the extension of a bijective ε-isometry and the existence of the Figiel operator of an isometric embedding between two unit spheres of Banach spaces. In the first part, we introduce and study the quasi-Figiel problem about a quasi-isometric embedding between unit spheres of two Banach spaces. Consequently, a quasi-anti-Lipschitz type inequality is obtained when the domain space is l∞n. Based on this quasi-anti-Lipschitz type inequality, the corresponding quasi-anti-Lipschitz type inequality is also obtained for an ε-isometric embedding defined on the unit sphere of an L∞,1+ space, i.e., a Banach space whose finite-dimensional subspaces can be any close to the finite-dimensional subspaces of l∞ in the sense of the Banach-Mazur distance. As an application of the quasi-anti-Lipschitz type inequality, we show that every bijective ε-isometry between the unit spheres of an L∞,1+ space and another Banach space can be extended to a bijective 5ε-isometry between their corresponding unit balls. In particular, this implies that every L∞,1+ space admits the Mazur-Ulam property. Furthermore, in this paper, some attempts are also made to generalize the classical Figiel's theorem to the local case with respect to an isometric embedding between unit spheres.

English

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Gordon, we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. This we achieve by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces is larger than the height of a closed boundary of the second space. Next we show that this estimate can be improved, if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of $\mathcal{C}(K, E)$ spaces.

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 \(X^*\).

On the stability of the Lp-curvature

It is known that the Lp-curvature of a smooth, strictly convex body in Rn is constant only for origin-centered balls when 1≠p>−n, and only for balls when p=1. If p=−n, then the L−n-curvature is constant only for origin-symmetric ellipsoids. We prove ‘local’ and ‘global’ stability versions of these results. For p≥1, we prove a global stability result: if the Lp-curvature is almost a constant, then the volume symmetric difference of K˜ and a translate of the unit ball B is almost zero. Here K˜ is the dilation of K with the same volume as the unit ball. For 0≤p<1, we prove a similar result in the class of origin-symmetric bodies in the L2-distance. In addition, for −n<p<0, we prove a local stability result: There is a neighborhood of the unit ball that any smooth, strictly convex body in this neighborhood with ‘almost’ constant Lp-curvature is ‘almost’ the unit ball. For p=−n, we prove a global stability result in R2 and a local stability result for n>2 in the Banach-Mazur distance.

Complexity of distances: Reductions of distances between metric and Banach spaces

We show that all the standard distances from metric geometry and functional analysis, such as Gromov-Hausdorff distance, Banach-Mazur distance, Kadets distance, Lipschitz distance, Net distance, and Hausdorff-Lipschitz distance have all the same complexity and are reducible to each other in a precisely defined way. This is done in terms of descriptive set theory and is a part of a larger research program initiated by the authors in \emph{Complexity of distances: Theory of generalized analytic equivalence relations}. The paper is however targeted also to specialists in metric geometry and geometry of Banach spaces.

On the topology of some hyperspaces of convex bodies associated to tensor norms

For every tuple d1,…,dl≥2, let Rd1⊗⋯⊗Rdl denote the tensor product of Rdi, i=1,…,l. Let us denote by B(d) the hyperspace of centrally symmetric convex bodies in Rd, d=d1⋯dl, endowed with the Hausdorff distance, and by B⊗(d1,…,dl) the subset of B(d) consisting of the convex bodies that are closed unit balls of reasonable crossnorms on Rd1⊗⋯⊗Rdl. It is known that B⊗(d1,…,dl) is a closed, contractible and locally compact subset of B(d). The hyperspace B⊗(d1,…,dl) is called the space of tensorial bodies. In this work we determine the homeomorphism type of B⊗(d1,…,dl). We show that even if B⊗(d1,…,dl) is not closed with respect to the Minkowski sum, it is an absolute retract homeomorphic to Q×Rp, where Q is the Hilbert cube and p=d1(d1+1)+⋯+dl(dl+1)2. Among other results, the relation between the Banach-Mazur compactum and the Banach-Mazur type compactum associated to B⊗(d1,…,dl) is examined.

Stability of the fixed point property for Hilbert spaces

We prove that if X is a Banach space whose Banach-Mazur distance to a Hilbert space is less than 2412+72+6(19+122), then X has the fixed point property for nonexpansive mappings. Also, a renorming of ℓ2 is constructed in order to show the limitations of the usual techniques.

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.

A solution to the fifth and the eighth Busemann-Petty problems in a small neighborhood of the Euclidean ball

We show that the fifth and the eighth Busemann-Petty problems have positive solutions for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance.

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.