Let
be a smooth algebraic hypersurface. Then the total curvature of is the "volume" of the Gauss map
.

The total curvature of the real Amoeba is then the volume of the image of
the Logarithmic Gauss map.

I will recall the definition of G. Mikhalkin, and the theorem of Mikhalkin-
Rullgard which characterize plane Simple Harnack curves by the fact that the
Amoeba has maximal area.

I will give a bound for the total curvature of the real Amoeba of a real plane
curve (in term of its Newton Polygon) and prove that this bound is reached
if and only if is a (smooth) simple Harnack curve.

If time , I will quote a recent result about total curvature of Real tropical
hypersurfaces.