On a class of Cheeger inequalities

Abstract

We study a general version of the Cheeger inequality by considering the shape functional F p,q (Ω) = λ 1/p p (Ω)/λ q (Ω) 1/q . The infimum and the supremum of F p,q are studied in the class of all domains Ω of R d and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for F p,q is discussed.