# Theory

Minkowski functionals have a rich background in integral geometry.

For sufficiently smooth bodies $K$, the Minkowski Functionals can be intuitively defined via (weighted) integrals over the volume or boundary of the body $K$.

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant. There are three independent Minkowski functionals in 2D, and four independent in 3D.

Minkowski vectors are closely related to the centers of mass in either solid or hollow bodies.

The second-rank tensors correspond the tensors of inertia, or they can be interpreted as the moment tensors of the distribution of the normals on the boundary. In 2D, there are four linear independent second-rank Minkowski tensors, in 3D there are six.

Each tensor of an anisotropic body describes both the preferred orientation and the amplitude or degree of the anisotropy. The latter can conveniently be described by anisotroy indices $\beta_\nu^{r,s}$ derived from second-rank Minkowski tensors in the Cartesian representation.

The irreducible representation, using spherical harmonics, provides scalar anisotropy indices (Spherical Minkowski) for higher-rank Minkowski tensors.