Hodge-Laplace eigenvalues of convex bodies.

We give upper and lower bounds of the first eigenvalue of the Hodge Laplacian acting on smooth $ 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$ [ABSTRACT FROM AUTHOR]