Let $K_1$ and $K_2$ two convex bodies in the euclideand space $\mathbb{E}^3$ (of dimesion 3):
Let's denote the $V$ as the volume and $S$ the surface area
I am looking for two thing(with reference):
1-The expression of the volume and the surface area of $K_1+K_2$ $$V(K_1+K_2) \quad \mbox{ and} \quad S(K_1+K_2)$$
2-the geometrical meaning of each terms in $V(K_1+K_2) \quad \mbox{ and} \quad S(K_1+K_2)$
For example, if $K_2$ is the euclidean unit ball $B$, by Steiner polynomial for the volume, we have $$V(K_1+B)=V(K_1)+S(K_1)+M+V(B)$$ $M $ here is the total mean curvature of $K_1$,the others terms are clear (surface area and volume )
in dimension 2, Minkowski sums act linearly on the perimeter: the perimeter of the sum equals the sum of perimeters (which is clearly not the case one dimension 3)