Let us begin with a discussion of the geometric question. If is an open set in (bounded or unbounded), let denote the lowest eigenvalue of in with Dirichlet boundary conditions. if is empty. Intuitively, if is small then must be large in some sense. One well known result in this direction is the Faber-Krahn inequality which states that among all domains with a given volume , the ball has the smallest . Thus,

where is the lowest eigenvalue of a ball of unit volume. This inequality clearly does not tell the whole story. If is small then must not only have a large volume, it must also be “fat” in some sense.

Let us place here a very beautiful result due to Lieb among other big contributions. This result was published in *Invent. Math.* during 1983. The proof relies upon the Rayleigh quotient and a very clever choice of a trial function for the variational characterization of . For the whole paper, we refer the reader to here.

Theorem. Let and be non-empty open sets in (), and and be the lowest eigenvalue of with Dirichlet boundary conditions. Let denote translated by . Let . Then there exists an such that.

If and are both bounded then there is an such that

.

Before proving the theorem, let us recall the so-called Rayleigh quotient. Precisely