# Min of a real-valued Fourier transform

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?

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Why would anyone downvote this question? – Igor Rivin Jul 6 '13 at 17:14
Note that $\mu$ can be supported on a NONCONVEX subset of $P$ of arbitrarily horrible topology, so the convexity of $P$ does not seem to help -- you need more hypotheses to get something sensible. – Igor Rivin Jul 6 '13 at 17:16
Thanks, Igor, for your comment and for understanding the question. I have a feeling that the answer is still quite meaningful, but perhaps we lack good tools to get there. For example, if $P:= [-1/2, 1/2]$, its Fourier transform is the usual Sinc function, which has a global min at around $\xi = 1.4$, with the definition of the transform given above. It also has infinitely many other local minima, but they are larger. Perhaps we should simply concentrate on the uniform measure? – Sinai Robins Jul 6 '13 at 19:22
I suspect that even with the uniform measure it might be impossible to say anyting useful in general. Suppose $d=2$ and let $P_0$ be the unit circle. Then $\hat\mu_{P_0}(\xi)$ is a radial function involving a Bessel function, which has a circle of global minima (if I did this right, $\hat\mu_{P_0}(\xi) = J_1(2\pi\left|\xi\right|) / (2\pi\left|\xi\right|)$, and the global minum is around $\left|\xi\right| \approx 0.817$). Now if we perturb $P_0$ a bit to some symmetric and convex $P$, there'll usually be a single pair $\pm\xi$ of global minima of $\hat\mu_P$ somewhere near that circle, ... – Noam D. Elkies Jul 6 '13 at 22:59
[interrupted by 600-character limit] ... but its position could jump unpredictably depending on the perturbation. – Noam D. Elkies Jul 6 '13 at 22:59