From affine Poincaré inequalities to affine spectral inequalities

Given a bounded open subset ? of R n , we establish the weak closure affine ball B p A ( ) = { f ? W 0 1 : E ? } with respect to functional introduced by Lutwak, Yang and Zhang in [46] as well its compactness L for any ? . These points use strongly celebrated Blaschke-Santaló inequality. As counterpart, develop basic theory -Rayleigh quotients domains, case, More specifically, -affine versions Poincaré inequality some their consequences. We introduce invariant -Laplace operator ? defining Euler-Lagrange equation minimization problem Rayleigh quotient. also study first eigenvalue ? which satisfies corresponding Faber-Krahn inequality, that is, is minimized (among sets equal volume) only when an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at present comparisons between classical eigenvalues, including result rigidity through characterization equality cases All inequalities obtained are stronger directly imply ones.