Hodge-Laplace eigenvalues of convex bodies

We give upper and lower bounds of the first eigenvalue of the Hodge Laplacian acting on smooth p-forms on a convex Euclidean domain for the absolute and relative boundary conditions. In particular, for the absolute conditions we show that it behaves like the squared inverse of the p-th longest principal axis of the ellipsoid of maximal volume included in the domain (the John ellipsoid). Using john's theorem, we then give a spectral geometric interpretation of the bounds and relate the eigenvalues with the largest volume of a p-dimensional section of the domain.