  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 $c\lambda^{-1/2}(\log\lambda)^{-(d-2)/2}$. This ball is centered
at a point at which the eigenfunction attains its maximum in absolute value
within the nodal domain.
