Explicit additive decomposition of norms on $\mathbb{R}^2$

TL;DR

This paper provides an explicit, elementary isometric embedding of any two-dimensional normed space into L1(0,1), with applications to moments of random vectors, inequalities, and statistical bounds.

Contribution

It offers a new explicit form for the isometric embedding of 2D normed spaces into L1, improving understanding and applications in probability and statistics.

Findings

Explicit embedding formula for 2D normed spaces into L1

Representation of moments of random vectors via characteristic functions

Partially improved inequalities related to Littlewood--Khinchin--Kahane

Abstract

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 Lata{\l}a and Oleszkiewicz; (iii) an extension of an inequality by Buja--Logan--Reeds--Shepp, arising from a statistical problem.