  Let $\mathcal{P}$ be an $n$-dimensional convex polytope and $\mathcal{S}$ be
a hypersurface in $\mathbb{R}^n$. This paper investigates potentials to
reconstruct $\mathcal{P}$ or at least to compute significant properties of
$\mathcal{P}$ if the modulus of the Fourier transform of $\mathcal{P}$ on
$\mathcal{S}$ with wave length $\lambda$, i.e., $|\int_{\mathcal{P}}
e^{-i\frac{1}{\lambda}\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}|$ for
$\mathbf{s}\in\mathcal{S}$, is given, $\lambda$ is sufficiently small and
$\mathcal{P}$ and $\mathcal{S}$ have some well-defined properties. The main
tool is an asymptotic formula for the Fourier transform of $\mathcal{P}$ with
wave length $\lambda$ when $\lambda \rightarrow 0$. The theory of X-ray
scattering of nanoparticles motivates this study since the modulus of the
Fourier transform of the reflected beam wave vectors are approximately
measurable in experiments.
