Taras Banakh
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.
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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.
In this note we correct a paper by D. Kang ("On the Mazur-Ulam theorem in non-Archimedean fuzzy anti-2-normed spaces", Filomat, 2017). The research in that paper applies to what the author calls strictly convex spaces. Nevertheless, we prove that this notion is void: there is no single space that satisfies the definition.
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 $\ell_{\infty}^m$.
We prove a local version of the Mazur-Ulam theorem.
In this paper, we establish the Mazur--Ulam theorem in the fuzzy strictly convex real normed spaces.
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.
In this paper, we generalize the Mazur--Ulam theorem in the fuzzy real n-normed strictly convex spaces.
Decompositions in direct sum of seminormed vector spaces and Mazur-Ulam theorem
该研究推广经典 Mazur–Ulam 定理，刻画半赋范实向量空间间保距映射具备线性性的充要条件。
In this article, we study the notions of $n$-isometries in non-Archimedean $n$-normed spaces over linear ordered non-Archimedean fields, and prove the Mazur-Ulam theorem in the spaces. Furthermore, we obtain some properties for $n$-isometries in non-Archimedean $n$-normed spaces.
Mazur-Ulam Theorem With Gromov-Hausdorff Distance
本文在 Gromov-Hausdorff 距离框架下，研究了 Banach 空间线性等距的充要条件，并给出了有限维情况下的简化证明与等距判据。
The classical Mazur--Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur--Ulam theorem in the non-Archimedean strictly convex normed spaces.
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, …
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.
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…
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…
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.
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.
A Probabilistic Generalization of the Mazur-Ulam Theorem
本文将马祖尔-乌拉姆定理泛化到概率空间，证明几乎处处保持距离的映射几乎处处与全局欧几里得等距映射一致。
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.
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 …
We prove that every unital C*-algebra $A$ has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space…
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。
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 …
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 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 $0.9$.
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。
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。
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…
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 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.
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。
The principal result in this note is a strengthened version of Kadison's transitivity theorem for unital JB$^*$-algebras, showing that for each minimal tripotent $e$ in the bidual, $\mathfrak{A}^{**}$, of a unital JB$^*$-algebra $\mathfrak{A}$, there exists a self-adjoint element $h$ in $\mathfrak{A}$ satisfying $e\leq \exp(ih)$, that is, $e$ is bounded by a unitary in the principal connected component of the unitary elements in $\mathfrak{A}$. This new result opens the way to attack new geometric results, for example, a Russo--Dye type theorem for maximal norm closed proper faces of the close…
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.
Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows: Let $ϕ:S^{2} \rightarrow S^{2}$ be a homeomorphism of order n and $λ\neq 1$ be an nth root of the unity, then for every complex valued continuous function $f$ on $S^{2}$ the function $\sum_{i=0}^{n-1} λ^{i}f(ϕ^{i}(x))$ must be vanished at some point of $S^{2}$. We give a generalization in term of action of compact groups. We also discuss about some noncommutative versions of the Borsuk- Ulam theorem
On some geometrization of compact metric spaces: A solution to the Banach-Ulam conjecture
提出基于Banach和Mycielski思想的紧度量空间几何化方法，证明了Banach-Ulam猜想：每个紧度量空间存在有限可加、合同不变的测度，并解决了Talagrand问题。
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.
The generalized Hyers--Ulam--Rassias stability of generalized derivations on unital Banach algebras into Banach bimodules is established.
On a problem of Mazur and Sternbach
本文针对 Mazur–Sternbach 第155号问题，借助度量几何的新进展给出另一种证明路径，并在局部等距假设下得到肯定结论。
By adopting the direct method and fixed point method, we prove that the Hyers-Ulam stability of the following additive-quadratic functional equation \begin{equation} f(x+y, z+w)+f(x-y, z-w)-2 f(x, z)-2 f(x, w)=0 \end{equation} in $β$-homogeneous $F$-spaces and quasi-Banach spaces. There are some differences that we consider the target space with the $β$-homogeneous norm and quasi-norm. Overcoming the $β$-homogeneous norm and quasi-norm bottlenecks, we get some new results.
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…
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。
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。
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。
Generalized gyrovector spaces revisited
本文修正了广义陀螺向量空间中Mazur-Ulam定理的一个证明。
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.
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.)
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。
We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.
In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then, as a special case linear isometries are characterized through a simple property of their range.
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 Certain Extremal Banach-Mazur Distances and Ader's Characterization of Distance Ellipsoids
本文证明三维空间中平行多面体与交叉多胞体是 Banach-Mazur 距离达到 $\sqrt{n}$ 的唯一极大化对象，并基于 Ader 的距离椭球刻画给出绕开体积概念的证明，同时建立平行四边形在平面问题中的唯一性与稳定性。
On the Scottish Book Problem 155 by Mazur and Sternbach
本文研究苏格兰书问题 155，证明当 Banach 空间 X 可分且映射仅为满射时，局部等距性质可推出全局等距。
In this paper, we establish the Hyers--Ulam--Rassias stability of ring homomorphisms and ring derivations on fuzzy Banach algebras.
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…
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