$\mathbb{R}^2$上范数的显式可加分解
Explicit additive decomposition of norms on $\mathbb{R}^2$
摘要：
Lindenstrauss的一个著名结果是，任何二维赋范空间都可以等距嵌入到$L_1(0,1)$中。我们提供了这样一个嵌入的显式形式。证明是基本的和独立的。给出了随机向量$X$的范数的矩的特征函数的显式表示和$X$的分布的傅里叶-拉普拉斯变换，以及Latatta和Oleszkiewicz得到的Littlewood-Khinchin-Kahane不等式的精确形式的显式和部分改进形式;（iii）一个由Buja-Logan-Reeds-Shepp提出的不等式的扩展，它源于一个统计问题。
英文摘要：
A well-known result by Lindenstrauss is that any two-dimensional normed space can be isometrically imbedded into $L_1(0,1)$. We provide an explicit form of a such an imbedding. The proof is elementary and self-contained. Applications are given concerning the following: (i) explicit representations of the moments of the norm of a random vector $X$ in terms of the characteristic function and the Fourier--Laplace transform of the distribution of $X$; (ii) an explicit and partially improved form of the exact version of the Littlewood--Khinchin--Kahane inequality obtained by Latała and Oleszkiewicz; (iii) an extension of an inequality by Buja--Logan--Reeds--Shepp, arising from a statistical problem.
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