Maximization of the Fourier-zero distance by the regular dodecagon

Determine whether the regular dodecagon of area \(\pi\) maximizes the functional \(\kappa(\Omega)=\operatorname{dist}(\mathcal{N}(\Omega),0)\) among all convex centrally symmetric planar domains of area \(\pi\).

Background

The paper studies the distance from the origin to the zero set of the Fourier transform of the characteristic function of a centrally symmetric convex body, under a fixed-area or fixed-volume constraint. In the planar case, the authors prove that regular polygons with at least twelve sides have a larger $\kappa$-value than the disk, and that among regular polygons the dodecagon has the largest value demonstrated by their monotonicity theorem.

These results identify the regular dodecagon as a candidate maximizer among all convex balanced planar domains, but the paper does not establish global maximality over arbitrary convex balanced domains. The question is therefore left explicitly unresolved.