Let $P$ be a compact, convex, symmetric, $d$-dimensional body in $\mathbb R^d$, and let $\mu$ be a (necessarily) symmetric probability measure on $P$, so that $\mu_P(x) = \mu_P(-x)$, for all $x \in \mathbb R^d$. As a consequence, the Fourier transform $\hat \mu_P(\xi) := \int_{\mathbb R^d} exp(2\pi i \langle \xi \cdot x \rangle d \mu$) is real-valued.

Are there methods to find a $\bf{global}$ minimum (it's ok if it's not necessarily unique) of the (real-valued) Fourier transform $\hat \mu_P(\xi)$?

Note. Of course, one can find conditions for local minima and then employ a brute-force comparison of all of them, but there are potentially infinitely many local minima. So I'm looking for something better than this. In other words, does the convexity of $P$ help us somehow?

somewherenear that circle, ... – Noam D. Elkies Jul 6 '13 at 22:59