Let 𝒫 and $P^{\prime }$ be 3-dimensional convex polytopes in $R^3$ and $S\subseteq R^3$ 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 $P^{\prime }$ coincide up to translation and/or reflection in a point if $\left|{\int }_P{e}^{-is\cdot x}\mathrm{dx}\right|=\left|{\int }_{P^{\prime }}{e}^{-is\cdot x}\mathrm{dx}\right|$ for all $s\in S$. 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.

中文翻译：

---

球体上的傅立叶变换的模确定3维凸多面体

让𝒫和 $P^{\prime }$ 是3维凸多面体 ${\mathrm{[R}}^3$ 和 $\mathrm{小号}\subseteq {\mathrm{[R}}^3$是开放集与球面的非空交集。作为更一般性结果的结果，证明𝒫和$P^{\prime }$ 如果某点重合到平移和/或反射 $\left|{\int }_P{Ë}^{--\mathrm{一世}s\cdot X}\mathrm{dx}\right|=\left|{\int }_{P^{\prime }}{Ë}^{--\mathrm{一世}s\cdot X}\mathrm{dx}\right|$ 对全部 $s\in \mathrm{小号}$。可以将其应用于晶体学领域，即是否可以通过其在Ewald球上的X射线衍射图谱的强度来唯一确定被建模为凸多面体的纳米颗粒的问题。