Konstantin M. Dyakonov
Let $Λ$ be a finite set of nonnegative integers, and let $\mathcal P(Λ)$ be the linear hull of the monomials $z^k$ with $k\inΛ$, viewed as a subspace of $L^1$ on the unit circle. We characterize the extreme and exposed points of the unit ball in $\mathcal P(Λ)$.
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Mutual singularity of Riesz products on the unit sphere
本文在$\mathbb{C}^n$单位球面上证明了标准与广义Riesz乘积的Peyrière相互奇异性定理的类似结果，并由此得到Zygmund二分性的相应版本。
The trace as an average over the unit sphere of a normed space with a 1-symmetric basis
我们将表示给定方阵的矩阵迹为$A$在欧氏球面上的数值积分的公式推广到具有1-对称基的有限维赋范空间的单位球面上。即使在$p\neq 2$的$\mathbb{R}^N$中的$\ell_p$ -范数的情况下，我们的结果也是新的。
Boundedness of projection operators and Cesàro means in weighted $L^p$ space on the unit sphere
对于单位球面上的权函数$\prod_{i=1}^{d+1}|x_i|^{2\k_i}$，得到了正交投影算子的局部估计，并证明了当$δ$低于临界指标时，Cesàro $(C,δ)$平均在加权$L^p$空间的收敛性.对于单位球和单形上相应的权函数也证明了类似的结果。
On Uniform Homeomorphisms of the Unit Spheres of Certain Banach Lattices
证明了：若X是具有弱单位的无穷维巴拿赫格，则存在概率空间（Omega，Sigma，mu）使得单位球面S（L_1（Omega，Sigma，mu））与单位球面S（X）一致同胚当且仅当X不一致包含l_{infty} ^n.
On Banach subalgebras of $\mathscr{H}^\infty$ consisting of lacunary Dirichlet series
本文研究由缺项狄利克雷级数构成的 $\mathscr{H}^\infty$ 的巴拿赫子代数 $\mathscr{H}^\infty_S$，刻画其成为含幺子代数的条件、乘子代数结构、单位群及无穷 Bass 稳定秩。
Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces
证明了有限维多面体巴拿赫空间的单位球面之间的一个上等距扩张到相应空间的一个线性等距。
We use the classical Fourier analysis to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained for totally geodesic Funk transforms on the sphere and the correpsonding lambda-cosine transforms.
Geodesic paths in the finite dimensional unit sphere under sup norm
在超范数下刻画了$n$维单位球面上的测地线.两点之间的测地线路径是连接两点的最短曲线。
On the plasticity of the unit spheres of $\ell_1$, $\ell_{\infty}$, $c$, and Hilbert spaces
本文研究了 $\ell_1$、$\ell_{\infty}$ 和 $c$ 的单位球的扩展-收缩可塑性，并证明了 Hilbert 空间单位球的强可塑性。
Let $Γ$ be an infinite set equipped with the discrete topology. We prove that the space $\ell_{\infty}(Γ),$ of all complex-valued bounded functions on $Γ$, satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of $\ell_{\infty}(Γ)$ onto the unit sphere of an arbitrary complex Banach space $X$ admits a unique extension to a surjective real linear isometry from $\ell_{\infty}(Γ)$ to $X$.
In this paper, we consider the CQ projection method and the shrinking projection method given by a finite family of nonexpansive mappings of the unit sphere of a real Hilbert space, and prove strong convergence theorems to common fixed points of the mappings.
We prove that every surjective isometry from the unit sphere of a rank-2 Cartan factor $C$ onto the unit sphere of a real Banach space $Y$, admits an extension to a surjective real linear isometry from $C$ onto $Y$. The conclusion also covers the case in which $C$ is a spin factor. This result closes an open problem and, combined with the conclusion in a previous paper, allows us to establish that every JBW$^*$-triple $M$ satisfies the Mazur--Ulam property, that is, every surjective isometry from its unit sphere onto the unit sphere of a arbitrary real Banach space $Y$ admits an extension to a…
We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two points at distance exactly two. Also, we study when the convex combinations of slices of the unit ball are relatively open or has non-empty relative interior for different topologies, studying the relationship between them and studying these properties for $L_{\infty}$-spaces and preduals of $L_1$-spaces.
Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}π\|x-y\|, \] where $d$ denotes the intrinsic metric on $S$.
Uniform homeomorphisms between spheres of unitarily invariant ideals
本文证明非交换版本的Odell-Schlumprecht定理，通过两类广义Mazur映射（基于p-凸化和量子相对熵最小化）建立酉不变理想球面与Hilbert空间球面间的均匀同胚，提供满足性质(H)的新例子。
A complete characterization of Radon planes whose unit spheres are regular polygons
本文从几何角度研究了多边形Radon平面的单位球结构，引入新的几何概念对其进行完整刻画，并证明了顶点数为4n的正多边形不能作为单位球，而4n+2个顶点的正多边形可以。
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.
Paley's theorem for Hankel matrices via the Schur test
单位圆上经典空间$H^1$中函数缺项系数的Paley定理等价于某些Hankel矩阵定义了非负整数的$\ell^2$上的有界算子。由于该陈述很容易简化为矩阵中的元素都是非负的情况，因此必须通过舒尔检验来证明。我们给这样的证明与有趣的模式中使用的向量在测试中，我们恢复最好的常数在主要情况下。利用相关的思想，将Paley乘子的特征从$H^1$重新证明为$H^2$ .
We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor.
Characterization of Supporting Functionals at Points of the Unit Sphere of Orlicz-Lorentz Spaces
该研究在不假设Orlicz函数为N-函数的条件下，完整刻画了Orlicz-Lorentz空间单位球面上任意点的支撑泛函，为该类空间的极值结构研究提供了基础。
Sharp Nash Inequalities on the unit sphere. The influence of symmetries
本文建立了$\mathbb{R}^{n+1}$的标准单位球面$\mathbb{S}^n$上Nash不等式的最佳常数，并给出了相应问题极值函数存在性的答案.研究了在群$G=O(k)\times O(m)$作用下数据不变的情况下的最佳常数问题，并找到了最佳常数。
Rigidity of unit spheres in finite-dimensional real normed spaces
本文证明有限维实赋范空间中单位球面等距蕴含线性等距，通过Plücker收缩体和零拉格朗日恒等式回答Kadets-Martín问题。
Let $f: S(E) \to S(B)$ be a surjective isometry between the unit spheres of two weakly compact JB$^*$-triples not containing direct summands of rank smaller than or equal to 3. Suppose $E$ has rank greater than or equal to 5. Applying techniques developed in JB$^*$-triple theory, we prove that $f$ admits an extension to a surjective real linear isometry $T: E\to B$. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C$^*$-algebras $A$ and $B$ (and in particular when $A=K(H)$ and $B=K(H')$) extends to a surjective real linear isometry from $A$…
Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$ denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded linear operators on $H$ and $K$, respectively. We prove that every surjective isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive answer to Tingley's problem in the setting of $B(H)$ spaces.
We study the problem of recovering an atomic measure on the unit 2-sphere $\mathbb{S}^2$ given finitely many moments with respect to spherical harmonics. The analysis relies on the formulation of this problem as an optimization problem on the space of bounded Borel measures on $\mathbb{S}^2$ as it was considered by Y. de Castro & F. Gamboa and E. Candés & C. Fernandez-Granda. We construct a dual certificate using a kernel given in an explicit form and make a concrete analysis of the interpolation problem. Numerical examples are provided and analyzed.
A variant of Tingley's problem for positive unit spheres of continuous functions that vanish at infinity
本文研究了正连续函数正单位球面的等距映射性质，证明了此类映射由空间同胚诱导，并扩展为连续函数空间的等距映射。
We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements $x,y\in A$, thereby answering a question of J. M. F. Castillo. In the case where $X$ contains an infinite-dimensional separable dual space or an unconditional basic sequence, the set $A$ may be chosen in a way that $\|x\pm y\| \geqslant 1+\varepsilon$ for some $\varepsilon > 0$ and distinct $x,y\in A$. Under additional structural properties of $X$, such as no…
We will introduce a new geometric constant GL(X) based on the constant H(X) proposed by Gao. We first further survey the constant H(X) and discuss some of the properties of this constant that have not yet been discovered. Next, we focus on a new constant GL(X) along with some of its basic properties. In addition, we show some relations between the well-known geometric constants and GL(X) through some inequalities. Finally, we characterize some generalized forms of the constant GL(X).
Given a C$^*$-algebra $A$, let $S(A^+)$ denote the set of those positive elements in the unit sphere of $A$. Let $H_1$, $H_2,$ $H_3$ and $H_4$ be complex Hilbert spaces, where $H_3$ and $H_4$ are infinite-dimensional and separable. In this note we prove a variant of Tingley's problem by showing that every surjective isometry $Δ: S(B(H_1)^+)\to S(B(H_2)^+)$ or (respectively, $Δ: S(K(H_3)^+)\to S(K(H_4)^+)$) admits a unique extension to a surjective complex linear isometry from $B(H_1)$ onto $B(H_2))$ (respectively, from $K(H_3)$ onto $B(H_4)$). This provides a positive answer to a conjecture po…
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