The inner radius of nodal domains in high dimensions

We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue $\lambda$ on a $d$-dimensional closed Riemannian manifold contains a ball of radius $c{\lambda }^{-1\mathrm{/}2}(\log \lambda {)}^{-(d-2)\mathrm{/}2}$. This ball is centered at a point at which the eigenfunction attains its maximum in absolute value within the nodal domain.