  We present a new proof of the sphere covering inequality in the spirit of
comparison geometry, and as a byproduct we find another sphere covering
inequality which can be viewed as the dual of the original one. We also prove
sphere covering inequalities on surfaces satisfying general isoperimetric
inequalities, and discuss their applications to elliptic equations with
exponential nonlinearities in dimension two. The approach in this paper
extends, improves, and unifies several inequalities about solutions of elliptic
equations with exponential nonlinearities.
