Benguria–Levitin–Parnovski conjecture

For $n\ge 2$, let $K\subset\mathbb{R}^n$ be a centrally symmetric convex body, meaning $K=-K$, and let $B\subset\mathbb{R}^n$ be a Euclidean ball satisfying $\lvert B\rvert=\lvert K\rvert$. Define $\widehat{\mathbf{1}_K}(\xi)=\int_K e^{-i\langle x,\xi\rangle}\,dx$ and $d(K)=\inf\{\lvert\xi\rvert:\widehat{\mathbf{1}_K}(\xi)=0\}$. The conjecture asserts that $d(K)\le d(B)$ for every such $K$; equivalently, among centrally symmetric convex bodies of fixed volume, the Euclidean ball maximizes the distance from the origin to the zero set of the Fourier transform of the characteristic function.