Beata Randrianantoanina
We prove that if $X$ is a real Banach space, with $\dim X\geq 3$, which contains a subspace of codimension 1 which is 1-complemented in $X$ and whose group of isometries is almost transitive then $X$ is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.
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On positive Banach-Mazur distance
本文研究了某些C(K)Banach空间的一侧正Banach-Mazur距离，基于[4]中的方法解决[2]中提出的问题。
On the Banach-Mazur Ellipse
本文通过新方法证明了规范平面中最小巴拿赫-马祖尔距离到单位圆的椭圆特征化，并揭示其与切比雪夫交替的关系。
We show that there are uncountably many geodesics between any two non-isometric $n$-dimensional normed spaces. We construct two explicit geodesics that can be used to describe all the points of the other geodesics.
In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $Δ$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a surjective real-linear isometry from $Z$ to $Y$. We prove that for every countable set $Γ$ with $\vert Γ\vert \geq 2$, the Banach space $\bigoplus_{γ\in Γ}^{c_0} X_γ$ satisfies the Mazur--Ulam property, whenever the Banach space $X_γ$ is strictly convex with dim$((X_γ)_{\mathbb{R}})\geq 2$ for every $γ$. Moreover we prove that the Banach space $C_0(K,X)$ satisfi…
In order to measure qualitative properties we introduce a notion of a type for arbitrary normed spaces which measures the worst possible growth of partial sums of sequences weakly converging to zero. The ideas can be traced back to Banach and Mazur who used this type to compare the so-called linear dimension of classical Banach spaces. As an application we compare the linear dimension and investigate isomorphy of some classical Banach spaces.
We establish variant Khintchine inequalities on normed spaces of Hanner type and cotype, in which the Rademacher distribution corresponding to classical Khintchine inequality is replaced by general symmetric distributions. The proof involves the $p$-barycenter and Birkhoff's ergodic theorem. More importantly, by employing these Khintchine inequalities, we get some lower bounds for Banach-Mazur distance between $l^p$-ball and a general centrally symmetric convex body.
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 …
We prove a normalized version of the restricted invertibility principle obtained by Spielman-Srivastava. Applying this result, we get a new proof of the proportional Dvoretzky-Rogers factorization theorem recovering the best current estimate. As a consequence, we also recover the best known estimate for the Banach-Mazur distance to the cube: the distance of every n-dimensional normed space from \ell_{\infty}^n is at most (2n)^(5/6). Finally, using tools from the work of Batson-Spielman-Srivastava, we give a new proof for a theorem of Kashin-Tzafriri on the norm of restricted matrices.
We study $C$-rich spaces, lush spaces, and $C$-extremely regular spaces concerning with the Mazur-Ulam property. We show that a uniform algebra and the real part of a uniform algebra with the supremum norm are $C$-rich spaces, hence lush spaces. We prove that a uniformly closed subalgebra of the algebra of complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is $C$-extremely regular provided that it separates the points of the underlying space and has no common zeros. In section 3 we exhibit descriptions on the Choquet bounday, the \vSilov bounday, …
A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property.
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.
We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces.
We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of $\ell_2^2$ belongs to the two-dimensional asymptotic structure and w…
Bounds for Banach-Mazur distances between some $C(K)$-spaces
研究了某些$C(K)$-空间之间Banach-Mazur距离的界，特别关注收敛序列空间之间的距离下界。
A Complex Analogue of Spencer's Six Standard Deviations Theorem and the Complex Banach-Mazur Distance
本文提出复数版Spencer六标准差定理猜想，并证明了n=2,3时成立，同时确定了复数ℓ_p²空间之间的Banach-Mazur距离。
We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (specially, separable $C$-rich subspaces of $C(K)$), and even the two-dimensional space with hexagonal norm. We obtain that the space $C(K,E)$ of the vector-valued continuous functions is a GL-space whenever $E$ is, and show that the GL-space is stable under $c_0$-, $l_1$- and $l_\infty$-sums. As an application, we prove that the Mazur-Ulam property holds for a larger class of Banach spaces, called local-GL-spaces, including all lush spaces and…
In this paper we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative BC-linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand-Mazur theorem for the bicomplex division Banach algebra.
Projection Constants and Banach-Mazur Distance in Codimension Two
本文证明余维数一或二且投影常数小于4/3的子空间满足特定Banach-Mazur距离上界，并猜想该上界对所有此类子空间成立。
On the Banach--Mazur radius of $\ell_\infty^n$
该研究证明了任意n维赋范空间与$\ell_\infty^n$的Banach--Mazur距离上界为$O(n^{2/3})$，推进了Banach--Mazur半径相关的几何泛函分析研究。
Extremal Banach-Mazur distance between a symmetric convex body and an arbitrary convex body on the plane
本文证明在平面上，若对称凸体与另一凸体的Banach-Mazur距离达到极值2，则该凸体必为三角形。
Equilateral dimension of the planar Banach--Mazur compactum
本文证明了在巴拿赫-马祖尔距离下，平面对称凸体中存在任意大的等边集，并展示了构造的最优性。
The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fréchet space has an infinite-dimensional separable quotient space, namely $\mathbb{R}^ω$. It is proved in this paper that every infinite-dimensional Fréchet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace whic…
The maximum of the Banach-Mazur distance $d_{BM}^M(X,\ell_\infty^n)$, where $X$ ranges over the set of all $n$-dimensional real Banach spaces, is difficult to compute. In fact, it is already not easy to get the maximum of $d_{BM}^M(\ell_p^n,\ell_\infty^n)$ for all $p\in [1,\infty]$. We prove that $d_{BM}^M(\ell_p^3,\ell_\infty^3)\leq 9/5,~\forall p\in[1,\infty]$. As an application, the following result related to Borsuk's partition problem in Banach spaces is obtained: any subset $A$ of $\ell_p^3$ having diameter $1$ is the union of $8$ subsets of $A$ whose diameters are at most $0.9$.
In this paper, we show that a Banach space $X$ has the Uniform Mazur Intersection Property (UMIP) if and only if every $f \in S(X^*)$ is uniformly w*-semidenting point of $B(X^*)$. We also prove analogous results for uniform w*-MIP.
On the Banach-Mazur Distance in Small Dimensions
本研究确定了三维和四维空间中立方体与其对偶之间的Banach-Mazur距离，分别等于9/5和2，证实了基于数值的猜想，并在二维中利用不对称常数构造了与所有对称凸体等距的凸体族。
We give a revision of the proof of a Mazur-Ulam theorem for generalized gyrovector spaces given in the paper "Generalized gyrovector spaces and a Mazur-Ulam theorem" published in Publ. Math. Debrecen, 87 (2015), 393--413.
Zonoids whose polars are zonoids: the Banach--Mazur distance need not tend to one
本文构造旋转高斯 zonoid，证明其极体也是 zonoid，并计算其到欧氏球的 Banach--Mazur 距离约为 1.10，从而否定该距离趋于 1 的猜想，并给出一个非希尔伯特空间及其对偶均等距嵌入 L1 的例子。
A short note on hereditary Mazur intersection property
本文通过Borwein和Fabian的例子证明继承性Mazur交集性质不蕴含弗雷歇平滑性。
We give a sufficient condition for a Banach space with which the homogeneous extension of a surjective isometry from the unit sphere of it onto another one is real-linear. The condition is satisfied by a uniform algebra and a certain extremely $C$-regular space of real-valued continuous functions.
We construct a continuous scalar-valued 2-polynomial, $W$, on the separable Hilbert space $l_2$ and an unbounded set $R\subset l_2$ such that (i) $W$ is bounded on an $ε$-neighbourhood of $R$; (ii) $W$ is unbounded on ${1/2} R$; (iii) consequently, $W$ does not factor through any bounded 1-polynomial on $l_2$ sending $R$ to a bounded set. This answers in the negative two 1935 questions asked by Mazur (problems 55 and 75 in the Scottish Book). The construction is valid both over $\R$ and $\C$. (In finite dimensions the questions were answered in the positive by Auerbach soon after being asked.)
K.\ S.\ Lau had shown that a reflexive Banach space has the Mazur Intersection Property (MIP) if and only if every closed bounded convex set is the closed convex hull of its farthest points. In this work, we show that in general this latter property is equivalent to a property stronger than the MIP. As corollaries, we recapture the result of Lau and characterize the w*-MIP in dual of RNP spaces.
Let $n\geq 3$, and let $B_1^n$ 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 $\mathcal G_m$ in ${\mathbb R}^n$ such that the Banach--Mazur distance $d(B_1^n,\mathcal G_m)$ satisfies $d(B_1^n,\mathcal G_m)\geq n^{5/9}\log^{-C}n$, where $C>0$ is a universal constant. The body $\mathcal G_m$ is obtained as a typical realization of a random polytope in ${\mathbb R}^n$ with $2m:=2n^C$ vertices (for a large constant $C$). The result improves upon an earlier estimate of S.Szar…
We introduce a flexible almost isometric version of the almost transitivity property of Banach spaces. With the help of this new notion we generalize to several directions a strong recent rotational characterization of Hilbert spaces due to Randrianantoanina. This characterization is a partial answer to the classical Banach-Mazur rotation problem.
We introduce the (T)-property, and prove that every Banach space with the (T)-property has the Mazur-Ulam property (briefly MUP). As its immediate applications, we obtain that almost-CL-spaces admitting a smooth point(specially, separable almost-CL-spaces) and a two-dimensional space whose unit sphere is a hexagon has the MUP. Furthermore, we discuss the stability of the spaces having the MUP by the $c_0$- and $\ell_1$-sums, and show that the space $C(K,X)$ of the vector-valued continuous functions has the the MUP, where $X$ is a separable almost-CL-space and $K$ is a compact metric space.
Exact Banach-Mazur distances of certain $\ell_p$-sums and cones
本文研究了有限维实normed空间的ℓ_p直接和与相关凸体锥构造的Banach-Mazur距离，通过最近的最优Banach-Mazur位置特性，推导出从X_1⊕_p…⊕_pX_k到欧几里得空间的距离公式，并探讨了单锥与双锥的几何性质及距离关系。
We study the metric entropy of the metric space $B_n$ of all n-dimensional Banach spaces (the so-called Banach-Mazur compactum) equipped with the Banach-Mazur (multiplicative) "distance" $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({B_n},d, 1+\varepsilon)$ is the smallest number of "balls" of "radius" $1+\varepsilon$ that cover $ B_n$, then for any $\varepsilon>0$ we have $$0<\liminf_{n\to \infty}n^{-1} \log\log N(B_n,d,1+\varepsilon)\le \limsup_{n\to \infty}n^{-1} \lo…
The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach's famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraïssé theory and Ramsey theory, and led to development of concept…
We prove that every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property. Thanks to the representation theory, we can identify commutative JB$^*$-triples as spaces of complex-valued continuous functions on a principal $\mathbb{T}$-bundle $L$ in the form $$C_0^\mathbb{T}(L):=\{a\in C_0(L):a(λt)=λa(t)\text{ for every } (λ,t)\in\mathbb{T}\times L\}.$$ We prove that every surjective isometry from the unit sphere of $C_0^\mathbb{T}(L)$ onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.
In this note we study how a concentration phenomenon can be transmitted from one measure $μ$ to a push-forward measure $ν$. In the first part, we push forward $μ$ by $π:supp(μ)\rightarrow \Ren$, where $πx=\frac{x}{\norm{x}_L}\norm{x}_K$, and obtain a concentration inequality in terms of the medians of the given norms (with respect to $μ$) and the Banach-Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the Banach-Mazur distance between $K$ and $L$. As a corollary we show that any normed probability space with good…
Banach-Mazur distances and basis constants of isotropic log-concave random spaces
本文研究了由等距对称随机多面体生成的随机赋范空间间的Banach-Mazur距离，证明了在广泛参数范围内，该距离下界为cn/ln(1+m/n)，并扩展了Gluskin定理至一般等距对数凹测度，揭示了Banach-Mazur紧致集的极值几何现象。
Let $(Ω,μ)$ be a $σ$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (Ω,μ)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $Δ: S(L^{\infty} (Ω,μ))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (Ω,μ)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry …
Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $Δ$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For …
A $5/8$ Lower Bound on the Banach--Mazur Distance to the Cross-Polytope
该研究针对高斯Gluskin多面体与交叉多面体的Banach-Mazur距离，在m=n³范围内证明了5/8的多项式下界，改进了前期结果，通过替换商事件的方法完成证明。
We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c_0. We call X elastic if for some K < infinity for every Banach space Y which embeds into X, the space Y is K-isomorphic to a subspace of X. We also prove that if X is a separable Banach space such that for some K < infinity every isomorph of X is K-elastic then X is finite dimensional.
Our main result states that, given a finite-dimensional vector space $E$, the pseudometric defined in the set of continuous quasinorms $\mathcal{Q}_0=\{\|\cdot\|:E\to\mathbb{R}\}$ as $$d(\|\cdot\|_X,\|\cdot\|_Y)=\min\{μ:\|\cdot\|_X \leqλ\|\cdot\|_Y\leqμ\|\cdot\|_X\text{ for some }λ\}$$ induces, in fact, a complete norm when we take the obvious quotient $\mathcal{Q}=\mathcal{Q}_0/\!\sim$ and define the appropriate operations on $\mathcal{Q}$. We finish the paper with a little explanation of how this space and the Banach-Mazur compactum are related.
This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces $\mcH_1$ and $\mcH_2$, such that $\mcH_1$ is a continuous dense embedding in $\mcB$ and $\mcB$ is a continuous dense embedding in $\mcH_2$. This is the best possible improvement of a theorem due to Mazur (see \cite{BA} and also \cite{PE1}). The second objective is show how $\mcH_2$ allows us to provide a positive answer to the Marcinkiewicz-basis (M-basis) problem.
We prove a very general theorem concerning the estimation of the expression $\|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}\|$ for different kinds of maps $T$ satisfying some general perurbated isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone theorem which improves the results of Jarosz and more recent results of D…
The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character $ω_1$ has a renorming with the Mazur intersection property. Combined with the previous result of Jim\' enez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP normability of Asplund spaces of density $ω_1$ is undecidable in ZFC.
Given an infinite set $Γ$, we prove that the space of complex null sequences $c_0(Γ)$ satisfies the Mazur-Ulam property, that is, for each Banach space $X$, every surjective isometry from the unit sphere of $c_0(Γ)$ onto the unit sphere of $X$ admits a (unique) extension to a surjective real linear isometry from $c_0(Γ)$ to $X$. We also prove that the same conclusion holds for the finite dimensional space $\ell_{\infty}^m$.
How many miles from $L_\infty$ to $\ell_\infty$?
本文研究了$L_\infty[0,1]$与$\ell_\infty$之间的Banach-Mazur距离，给出了上下界。
Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier $\mathscr{L}_\infty$-Spaces
本文解决马祖尔问题，证明满足特定条件的L_∞空间存在c_0商，通过构造性方法和反例验证假设的必要性。
We study spaces with directionally asymptotically controlled ellipsoids approximating the unit ball in finite-dimensions. These ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids with a given functional of the dual space. The term asymptotical refers to the fact that we take '$\limsup$' over finite-dimensional subspaces. This leads to some isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem…
A generalization of Problem 73 of Mazur and Orlicz in the Scottish Book was introduced from L. Harris. The exact value of the constant that appears there is known when complex normed linear spaces are considered. In this paper, we give estimates in the case of an arbitrary real normed linear space and a real $\ell_p$ space. Moreover, if $F(x)$ is a power series, $ρ$ its radius of uniform convergence and $ρ_{\substack{A}}$ its radius of analyticity, we prove that $ρ_{\substack{A}}\geqρ/\sqrt{2}$ and give some respective results for the $n$th Fréchet derivative of $F(x)$.
Duality of uncomplemented null-space and non-closed range and its impact on the regularization of Mazur-type operators
该研究揭示了巴拿赫空间中无补零空间与非闭值域导致的不适定性的对偶关系，将其应用于Mazur型算子，提出可通过转移不适定性实现Tikhonov正则化的应用，为新正则化方法开发提供替代路径。
We prove that von Neumann algebras and separable nuclear $C^*$-algebras are stable for the Banach-Mazur cb-distance. A technical step is to show that unital almost completely isometric maps between $C^*$-algebras are almost multiplicative and almost selfadjoint. Also as an intermediate result, we compare the Banach-Mazur cb-distance and the Kadison-Kastler distance. Finally, we show that if two $C^*$-algebras are close enough for the cb-distance, then they have at most the same length.
Motivated by some recent twaddles on Mazur rotations problem, we study the "dynamics" of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on the metric aspects of the "action" of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.
On the expansiveness of coarse maps between Banach spaces and geometry preservation
该研究提出巴拿赫空间间新的可嵌入性概念，证明其严格弱于粗可嵌入性，借助度量余型技术研究余型保持与ℓ_p空间嵌入，结合Kalton非线性不变量对比，还探讨了ℓ_∞嵌入问题。
This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach space $X$ enjoys this property if and only if $C(K,X)$ enjoys this property. We also show the same result holds for $L_\infty(μ,X)$ and $L_1(μ,X)$.
The skew generalized von Neumann-Jordan type constant in Banach spaces
本文基于von Neumann-Jordan型常数C(X)定义了新的斜广义常数Cp(λ,μ,X)，研究了其基本性质及与其他常数的关系，并利用它与弱正交系数ω(X)的联系建立了正规结构的充分条件。
In this paper we investigate real convex-transitive Banach spaces X, which admit a 1-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. The results obtained can be regarded as partial answers to the well-known Banach-Mazur rotation problem, as well as to a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.
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