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

Abstract

Abstract Let 𝒫 and P ′ \mathcal{P}^{\prime} be 3-dimensional convex polytopes in R 3 \mathbb{R}^{3} and S ⊆ R 3 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 𝒫 and P ′ \mathcal{P}^{\prime} coincide up to translation and/or reflection in a point if | ∫ P e - i ⁢ s ⋅ x ⁢ dx | = | ∫ P ′ e - i ⁢ s ⋅ x ⁢ dx | \bigl{\lvert}\int_{\mathcal{P}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert}=\bigl{\lvert}\int_{\mathcal{P}^{\prime}}e^{-i\mathbf{s}\cdot\mathbf{x}}\,\mathbf{dx}\bigr{\rvert} for all s ∈ S \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.