  Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH)
manifold with a conformal compactification whose conformal infinity is
($\partial$M, [$\gamma$]). We will first observe that Ch(M, g) $\le$ n, where
Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci
curvature of M is bounded from below by --n and its scalar curvature approaches
--n(n+1) fast enough at infinity, then Ch(M, g) = n if and only Y($\partial$M,
[$\gamma$]) $\ge$ 0, where Y($\partial$M, [$\gamma$]) denotes the Yamabe
invariant of the conformal infinity. This gives an answer to a question raised
by J. Lee [L].
