Mazur Distance Research Articles

Stability constants of the weak⁎ fixed point property for the space ℓ1

The main aim of the paper is to study some quantitative aspects of the stability of the weak⁎ fixed point property for nonexpansive mappings in ℓ1 (shortly, w⁎-fpp). We focus on two complementary approaches to this topic. First, given a predual X of ℓ1 such that the σ(ℓ1,X)-fpp holds, we precisely establish how far, with respect to the Banach–Mazur distance, we can move from X without losing the w⁎-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in ℓ1 containing all σ(ℓ1,X)-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the w⁎-fpp in the restricted framework of preduals of ℓ1. Namely, we show that every predual X of ℓ1 with a distance from c0 strictly less than 3, induces a weak⁎ topology on ℓ1 such that the σ(ℓ1,X)-fpp holds.

Iterations of the projection body operator and a remark on Petty's conjectured projection inequality

We prove that if a convex body has an absolutely continuous surface area measure, whose density is sufficiently close to a constant function, then the sequence {ΠmK} of convex bodies converges to the ball with respect to the Banach–Mazur distance, as m→∞. Here, Π denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.

On strong property (T) and fixed point properties for Lie groups

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional subspaces is bounded above by a power of k strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank depending on X has strong property (T) of Lafforgue with respect to X. As a consequence, we obtain that every continuous affine isometric action of such a high rank group (or a lattice in such a group) on X has a fixed point. This result corroborates a conjecture of Bader, Furman, Gelander and Monod. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. Without appealing to strong property (T), we prove that given a Banach space X as above, every special linear group of sufficiently large rank satisfies the following property: every quasi-1-cocycle with values in an isometric representation on X is bounded.

Approximation properties for noncommutative Lp -spaces of high rank lattices and nonembeddability of expanders

Abstract This article contains two rigidity type results for SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} for large n that share the same proof. Firstly, we prove that for every p ∈ [ 1 , ∞ ] {p\in[1,\infty]} different from 2, the noncommutative L p {L^{p}} -space associated with SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} does not have the completely bounded approximation property for sufficiently large n depending on p. The second result concerns the coarse embeddability of expander families constructed from SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} . Let X be a Banach space and suppose that there exist β < 1 2 {\beta<\frac{1}{2}} and C > 0 {C>0} such that the Banach–Mazur distance to a Hilbert space of all k-dimensional subspaces of X is bounded above by C ⁢ k β {Ck^{\beta}} . Then the expander family constructed from SL ⁢ ( n , ℤ ) {\mathrm{SL}(n,\mathbb{Z})} does not coarsely embed into X for sufficiently large n depending on X. More generally, we prove that both results hold for lattices in connected simple real Lie groups with sufficiently high real rank.

The fixed point property for some generalized nonexpansive mappings and renormings

In 2008, T. Suzuki [33] defined one of most relevant extensions of the notion of nonexpansivity. This notion was extended in [15] defining the, so called, Cλ-mappings. In the last years, some papers have appeared trying to extend the most important results about existence of fixed points for nonexpansive mappings to this wider class of mappings (see, for instance, [2,5,9,10,15]). In this paper we continue this project proving that Banach spaces with extended unconditional bases satisfy the fixed point property for Cλ-mappings if the unconditional constant of the basis is small enough. We also begin a new project on renorming with the fixed point property for Cλ-mappings, proving that any separable Banach space can be renormed to satisfy the fixed point property for Cλ-mappings in a stable sense, that is, the space with the new norm satisfies the fixed point property which, in addition, is inherited by those isomorphic spaces which are close to it in the sense of the Banach–Mazur distance.

On the Local Convexity of Intersection Bodies of Revolution

AbstractOne of the fundamental results in convex geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach–Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

Characterizations of [formula omitted] and [formula omitted], and their stabilities

We characterize l∞n and l1n by introducing two geometric constants in n-dimensional Banach spaces. We also prove that these characterizations are stable with respect to the Banach–Mazur distance.

Distances between Hilbertian operator spaces

We compute the completely bounded Banach–Mazur distance between different finite-dimensional homogeneous Hilbertian operator spaces.

An approximate version of the Jordan von Neumann theorem for finite-dimensional real normed spaces

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite-dimensional normed vector spaces over , we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant for yields a small Banach–Mazur distance with , . Finally, we examine how this estimate worsens as increases, with the conclusion that grows quadratically with .

Notes on Schneider’s stability estimates for convex sets

In 2009 Schneider obtained stability estimates in terms of the Banach–Mazur distance for several geometric inequalities for convex bodies in an n-dimensional normed space \({\mathbb{E}^n}\). A unique feature of his approach is to express fundamental geometric quantities in terms of a single function \({\rho:\mathfrak{B} \times \mathfrak{B} \to \mathbb{R}}\) defined on the family of all convex bodies \({\mathfrak{B}}\) in \({\mathbb{E}^n}\). In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric dD on \({\mathfrak{B}}\) inducing, from our point of view, a finer topology than Banach–Mazur’s dBM. Further, dD induces a metric on the quotient \({\mathfrak{B}/{\rm Dil}^+}\) of \({\mathfrak{B}}\) by the relation of positive dilatation (homothety). Unlike its compact Banach–Mazur counterpart, dD is only “boundedly compact,” in particular, complete and locally compact. The general linear group \({{\rm GL}(\mathbb{E}^n)}\) acts on \({\mathfrak{B}/{\rm Dil}^+}\) by isometries with respect to dD, and the orbit space is naturally identified with the Banach–Mazur compactum \({\mathfrak{B}/{\rm Aff}}\) via the natural projection \({\pi:\mathfrak{B}/{\rm Dil}^+\to\mathfrak{B}/{\rm Aff}}\), where Aff is the affine group of \({\mathbb{E}^n}\). The metric dD has the advantage that many geometric quantities are explicitly computable. We show that dD provides a simpler and more fitting environment for the study of stability; in particular, all the estimates of Schneider turn out to be valid with dBM replaced by dD.

Minimal volume product near Hanner polytopes

Mahlerʼs conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in a fixed dimension. It is known that every Hanner polytope has the same volume product as the cube or the cross-polytope. In this paper we prove that every Hanner polytope is a strict local minimizer for the volume product in the class of symmetric convex bodies endowed with the Banach–Mazur distance.

Isomorphic properties of intersection bodies

We study isomorphic properties of two generalizations of intersection bodies – the class I k n of k-intersection bodies in R n and the class B P k n of generalized k-intersection bodies in R n . In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in R n and 1 ≤ k ≤ n − 1 then the outer volume ratio distance from K to the class B P k n can be estimated by o.v.r. ( K , B P k n ) : = i n f { ( | C | | K | ) 1 n : C ∈ B P k n , K ⊆ C } ≤ c n k l o g e n k , where c > 0 is an absolute constant. Next we prove that if K is a symmetric convex body in R n , 1 ≤ k ≤ n − 1 and its k-intersection body I k ( K ) exists and is convex, then d B M ( I k ( K ) , B 2 n ) ≤ c ( k ) , where c ( k ) is a constant depending only on k, d B M is the Banach–Mazur distance, and B 2 n is the unit Euclidean ball in R n . This generalizes a well-known result of Hensley and Borell. We conclude the paper with volumetric estimates for k-intersection bodies.

The unit ball is an attractor of the intersection body operator

The intersection body of a ball is again a ball. So, the unit ball B d ⊂ R d is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach–Mazur distance. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to B d in Banach–Mazur distance converge to B d in Banach–Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of B d . We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.

On Nearly Equilateral Simplices and Nearly l∞ Spaces

AbstractBy d(X, Y) we denote the (multiplicative) Banach–Mazur distance between two normed spaces X and Y. Let X be an n-dimensional normed space with d(X, ) ≤ 2, where stands for ℝn endowed with the norm ║(x1, … , xn)║∞ := max﹛|x1|, … , |xn|﹜. Then every metric space (S, ρ) of cardinality n + 1 with norm ρ satisfying the condition maxD/minD ≤ 2/ d(X, ) for D := ﹛ρ(a, b) : a, b ∈ S, a ≠ b﹜ can be isometrically embedded into X.

The Banach–Mazur Distance to the Cube in Low Dimensions

We show that the Banach–Mazur distance from any centrally symmetric convex body in ℝn to the n-dimensional cube is at most $$\sqrt{n^2-2n+2+\frac{2}{\sqrt{n+2}-1}},$$ which improves previously known estimates for “small” n≥3. (For large n, asymptotically better bounds are known; in the asymmetric case, exact bounds are known.) The proof of our estimate uses an idea of Lassak and the existence of two nearly orthogonal contact points in John’s decomposition of the identity. Our estimate on such contact points is closely connected to a well-known estimate of Gerzon on equiangular systems of lines.

Stability of the reverse Blaschke–Santaló inequality for zonoids and applications

An important GL ( n ) invariant functional of centred (origin symmetric) convex bodies that has received particular attention is the volume product. For a centred convex body A ⊂ R n it is defined by P ( A ) : = | A | ⋅ | A ∗ | , where | ⋅ | denotes volume and A ∗ is the polar body of A. If A is a centred zonoid, then it is known that P ( A ) ⩾ P ( C n ) , where C n is a centred affine cube, i.e. a Minkowski sum of n linearly independent centred segments. Equality holds in the class of centred zonoids if and only if A is a centred affine cube. Here we sharpen this uniqueness statement in terms of a stability result by showing in a quantitative form that the Banach–Mazur distance of a centred zonoid A from a centred affine cube is small if P ( A ) is close to P ( C n ) . This result is then applied to strengthen a uniqueness result in stochastic geometry.

Allometry constants of finite-dimensional spaces: theory and computations

We describe the computations of some intrinsic constants associated to an n-dimensional normed space $${\mathcal{V}}$$, namely the N-th “allometry” constants $$\kappa_\infty^N(\mathcal{V}) := \inf \{\|T\| \cdot \|T'\|, \quad T\,:\,\ell_\infty^N \to \mathcal{V}, \; \; T': \mathcal{V} \to \ell_\infty^N, \; \;TT'={\rm Id}_{\mathcal{V}}\}.$$ These are related to Banach–Mazur distances and to several types of projection constants. We also present the results of our computations for some low-dimensional spaces such as sequence spaces, polynomial spaces, and polygonal spaces. An eye is kept on the optimal operators T and T′, or equivalently, in the case N = n, on the best conditioned bases. In particular, we uncover that the best conditioned bases of quadratic polynomials are not symmetric, and that the Lagrange bases at equidistant nodes are best conditioned in the spaces of trigonometric polynomials of degree at most one and two.

Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

In (Gluskin, Litvak in Geom. Dedicate 90:45–48, [2002]) it was shown that a polytope with few vertices is far from being symmetric in the Banach–Mazur distance. More precisely, it was shown that Banach–Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, in fact it satisfies the same lower bound as the Banach–Mazur distance. In a sense it shows the following phenomenon: if a convex polytope with small number of vertices is as close to a symmetric body as it can be, then most of its vertices are as bad as the worst one. We apply our results to provide a lower estimate on the vertex index of a symmetric convex body, which was recently introduced in (Bezdek, Litvak in Adv. Math. 215:626–641, [2007]). Furthermore, we give the affirmative answer to a conjecture by Bezdek (Period. Math. Hung. 53:59–69, [2006]) on the quantitative illumination problem.

Banach–Mazur distance of planar convex bodies

We prove that the Banach–Mazur distance between arbitrary two planar convex bodies is at most 3. We also discuss a related distance which depends on affine-regular hexagons inscribed in the bodies.

On convex perturbations with a bounded isotropic constant

Let $$ K \subset {\user2{\mathbb{R}}}^{n} $$ be a convex body and ɛ > 0. We prove the existence of another convex body $$ K' \subset {\user2{\mathbb{R}}}^{n} $$ , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than $$ c \mathord{\left/ {\vphantom {c {{\sqrt \varepsilon }}}} \right. \kern-\nulldelimiterspace} {{\sqrt \varepsilon }} $$ , where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain.