Projection functions, area measures and the Alesker-Fourier transform

TL;DR

This paper introduces j-projection bodies as a generalization of projection bodies, characterizes them using Fourier analysis and area measures, and explores their properties with new examples.

Contribution

It provides the first Fourier analytic characterization of j-projection bodies and links them to Alesker's Fourier transform on valuations.

Findings

Characterization of j-projection bodies via Fourier analysis

Establishment of properties of j-projection bodies

Construction of new non-trivial examples

Abstract

Dual to Koldobsky's notion of j-intersection bodies, the class of j-projection bodies is introduced, generalizing Minkowski's classical notion of projection bodies of convex bodies. A Fourier analytic characterization of j-projection bodies in terms of their area measures of order j is obtained. In turn, this yields an equivalent characterization of j-projection bodies involving Alesker's Fourier type transform on translation invariant smooth spherical valuations. As applications of these results, several basic properties of j-projection bodies are established and new non-trivial examples are constructed.