The classical Brunn-Minkowski theory is based on the Minkowski addition of convex bodies, combined with the notion of volume. Building on the ideas presented earlier, we undertake a more in-depth and systematic study of the extension of this theory to \(\mathrm {C}^2\) -hedgehogs. After recalling the necessary basic tools and their geometric interpretations, we give a partial extension of the Alexandrov-Fenchel inequality to hedgehogs. We then consider the extension to hedgehogs of particular cases of the Alexandrov-Fenchel inequality (isoperimetric inequalities, quadratic Minkowskian inequalities, Brunn-Minkowski inequalities, etc), as well as some of their geometric consequences. In some cases, the classical inequalities for convex bodies extend to hedgehogs without modification by simply replacing the geometric quantities involved by their algebraic versions (as is the case for the isoperimetric inequality in the plane). But, in most cases, an additional condition is necessary. We give a stability estimate for the Alexandrov-Fenchel inequality. Finally, we give an application of hedgehogs to the study of the Blaschke diagram.