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

## 2D

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant (for compact bodies it is given by the number of components minus the number of holes).

Area

Perimeter

Euler characteristic

with

= curvature

## 3D

The scalar functionals can be interpreted as area, perimeter, or the Euler characteristic, which is a topological constant (for compact bodies it is given by the number of components minus the number of rings plus the number of cavities).

Volume

Surface area

Integrated mean curvature

(Mean width for convex bodies)

Euler Characteristic

with

= principal curvature