# 3D Functionals

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

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant. The vectors are closely related to the centers of mass in either solid or hollow bodies. Accordingly, 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.

### Minkowski Functionals Volume Surface area Integrated mean curvature
(Mean width for convex bodies) Euler Characteristic with   = principal curvature

## Cartesian representation (Minkowski Tensors)

Using the position vector and the normal vector on the boundary, the Minkowski Vectors can be defined in the Cartesian representation.

The second-rank Minkowski tensors are defined using the symmetric tensor product .

### Minkowski Vectors    ### Minkowski Tensors      Figures are under construction

## Irreducible representation (Spherical Minkowski Tensors)

under construction