An isoperimetric problem for Fourier zeros of centrally symmetric convex bodies

Given a centrally symmetric convex body, consider the zero set of the Fourier transform of its characteristic function. We study how large the distance from this set to the origin can be when the volume of the body is fixed. A 2009 conjecture of Benguria, Levitin, and Parnovski asserts that the maximum is attained by a Euclidean ball. We disprove it in all dimensions greater than one. In the plane, every regular centrally symmetric polygon with at least twelve sides outperforms the disk, albeit by a tiny margin, and the regular dodecagon is best among them. In dimensions three and higher, no maximiser exists.