Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$.  It is convenient to think of each of these points as lying in $\mathbb{C}$, and so a polygonal curve with n vertices can be thought of as a point in $\mathbb{C}^n$.

Now consider the discrete Fourier transform of this polygon:

\begin{equation}
X_k = \sum_{j=0}^{n-1}x_j \cdot e^{-2\pi ijk/n}, k \in \mathbb{Z}.
\end{equation}

Is it possible to determine directly from these Fourier coefficients $X_k$ whether our original polygonal curve is convex or not?  or whether it is simple (non-self-intersecting) or not?