Minkowski scalars are robust and intuitive shape measures. For sufficiently smooth bodies , the Minkowski Scalars are just (weighted) integrals over the volume or boundary of the body . There are three linearly independent Minkowski scalars in two dimensions, and four in three dimensions. Minkowski scalars are also known as intrinsic volumes.

Body

Area

Perimeter

Euler characteristic

where is the curvature of the contour at point . The Euler characteristic is given by the number of components minus the number of cavities:

### Hadwiger’s characterization theorem

Hadwiger’s characterization theorem [1] states that any additive, continuous, and motion-invariant functional of a convex body can be expressed as a linear combination of the Minkowski functionals of the body :

(1)

Here, are constants independent of . The shape information about is completely contained in the , which are , , and . In this sense, the Minkowski scalars are the relevant shape indices for physical functionals with the additivity property.