Multiplicative convolution with symmetries in Euclidean space and on the sphere

TL;DR

The paper studies multiplicative convolution on $\mathbb{R}^n$ and the spherical product on $S^{n-1}$ under groups of coordinate reflections, aiming to characterise universal signed measures—those for which $\mu * \nu = 0$ (or $\mu \circledast \nu = 0$) forces $\nu=0$. It develops a comprehensive framework based on coordinate decompositions, projections, and symmetry operators, introduces a lifting map to connect Euclidean and spherical settings, and proves precise universality criteria in both the Euclidean and spherical contexts, including special cases like origin-symmetric and unconditional measures. The results extend convex-geometry analyses of generalised zonoids and provide tools to decompose measures into symmetry components with explicit moment conditions that control injectivity. The lifting and symmetry-decomposition machinery yield a unified approach to universality, enabling transfer of results between $\mathbb{R}^n$ and $S^{n-1}$ and enabling generalisations of Molchanov–Nagel type theorems for convex bodies.

Abstract

Multiplicative convolution $μ\ast ν$ of two finite signed measures $μ$ and $ν$ on $\mathbb{R}^n$ and a related product $μ\circledast ν$ on the sphere $S^{n-1}$ are studied. For fixed $μ$ the injectivity in $ν$ of both operations is characterised given an arbitrary group of reflections along the coordinate axes. The results for the sphere yield generalised versions of the theorems in Molchanov and Nagel (2021) about convex bodies.

Theorems & Definitions (240)

• Remark 2.1
• Example 2.2
• Proposition 2.3
• Remark 2.4
• Proposition 2.5
• Remark 2.6
• Proposition 2.7
• proof
• Remark 2.8
• Proposition 2.9