Estimates on the Neumann and Steklov principal eigenvalues of collapsing domains

We investigate the relationship between the Neumann and Steklov principal eigenvalues emerging from the study of collapsing convex domains in ${\mathbb{R}}^2$. Such a relationship allows us to give a partial proof of a conjecture concerning estimates of the ratio of the former to the latter: we show that thinning triangles maximize the ratio among convex thinning sets, while thinning rectangles minimize the ratio among convex thinning with some symmetry property.