The amoeba of a holomorphic function $f$ is, by definition, the
image in $\mathbf{R}^n$ of the zero locus of $f$ under the simple
mapping that takes each coordinate to the logarithm of its
modulus. The terminology was introduced in the 1990s by the famous
(biologist and) mathematician Israel Gelfand and his coauthors
Kapranov and Zelevinsky (GKZ). In this paper we study a natural
convex potential function $N_f$
with the property that its
Monge-Ampére mass is concentrated to the amoeba of
$f$
We
obtain results of two kinds; by approximating $N_f$ with a
piecewise linear function, we get striking combinatorial
information regarding the amoeba and the Newton polytope of
$f$ ;
by computing the Monge-Ampére measure, we find sharp bounds for
the area of amoebas in $\mathbf{R}^n$ . We also consider systems of
functions $f_{1},\dots,f_{n}$
and prove a local version of the
classical Bernstein theorem on the number of roots of systems of
algebraic equations.