  The a-maximization technique proposed by Intriligator and Wecht allows us to
determine the exact R-charges and scaling dimensions of the chiral operators of
four-dimensional superconformal field theories. The problem of existence and
uniqueness of the solution, however, has not been addressed in general setting.
In this paper, it is shown that the a-function has always a unique critical
point which is also a global maximum for a large class of quiver gauge theories
specified by toric diagrams. Our proof is based on the observation that the
a-function is given by the volume of a three dimensional polytope called
"zonotope", and the uniqueness essentially follows from Brunn-Minkowski
inequality for the volume of convex bodies. We also show a universal upper
bound for the exact R-charges, and the monotonicity of a-function in the sense
that a-function decreases whenever the toric diagram shrinks. The relationship
between a-maximization and volume-minimization is also discussed.
