1 Total curvature

Let $X\subset\mathbb{R}^{n+1}$ be a smooth algebraic hypersurface. Then the total curvature of $X$ is the "volume" of the Gauss map $g:X\rightarrow \mathbb{R}P^n$.

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

2 Simple Harnack curves

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.

3 Total Curvature of the Real Amoeba

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

4 Total curvature of tropical hypersurfaces

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