The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes

TL;DR

This paper shows that the shape of a 3D convex polytope can be uniquely identified by the modulus of its Fourier transform on a sphere, with applications to determining nanoparticle shapes from X-ray diffraction data.

Contribution

It proves that convex polytopes are uniquely determined up to translation and reflection by the modulus of their Fourier transform on a sphere, extending previous results.

Findings

Convex polytopes are uniquely identified by Fourier modulus on a sphere.

Application to nanoparticle shape determination from X-ray diffraction.

Provides a theoretical foundation for shape reconstruction in crystallography.

Abstract

Let $\mathcal{P}$ and $\mathcal{P}'$ be $3$-dimensional convex polytopes in $\mathbb{R}^3$ and $S \subseteq \mathbb{R}^3$ be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that $\mathcal{P}$ and $\mathcal{P}'$ coincide up to translation and/or reflection in a point if $|\int_{\mathcal{P}} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}| = |\int_{\mathcal{P}'} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}|$ for all $\mathbf{s} \in S$. This can be applied to the field of crystallography regarding the question whether a nanoparticle modelled as a convex polytope is uniquely determined by the intensities of its X-ray diffraction pattern on the Ewald sphere.