The diameter of the first nodal line of a convex domains

The main goal of this paper is to prove that the first nodal line for the Dirichlet problem in a convex planar domain has diameter less than an absolute constant times the inradius of the domain. More precisely, we locate the nodal line, to within a distance comparable to the inradius, near the zero of an ordinary differential equation, which is associated to the domain in a natural way. We also derive estimates for the first and second eigenvalues in terms of the corresponding eigenvalues of the ordinary differential equation and construct an approximate first eigenfunction. Two examples, a rectangle and a circular sector, illustrate the two extreme possibilities for the location of the nodal line. For the rectangle R = { (x, y) o 1, we have the second eigenfunction