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

Abstract

Let $\mathcal{P}$ and ${\mathcal{P}}^{\prime }$ 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 ${\mathcal{P}}^{\prime }$ coincide up to translation and/or reflection in a point if $|{\int }_{\mathcal{P}}{e}^{-i\mathbf{s}\cdot \mathbf{x}}\mathbf{dx}|=|{\int }_{{\mathcal{P}}^{\prime }}{e}^{-i\mathbf{s}\cdot \mathbf{x}}\mathbf{dx}|$ 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.

12 pages, 1 figure