  We present the proof of several inequalities using the technique introduced
by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we
give a new and simple proof of a lower bound of Berestycki, Nirenberg, and
Varadhan concerning the principal eigenvalue of an elliptic operator with
bounded measurable coefficients. The rest of the paper is a survey on the
proofs of several isoperimetric and Sobolev inequalities using the ABP
technique. This includes new proofs of the classical isoperimetric inequality,
the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric
inequality in convex cones. For this last inequality, the new proof was
recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work
where we also prove new Sobolev inequalities with weights which came up
studying an open question raised by Haim Brezis.
