Minkowski scalars, also known as intrinsic volumes, are robust and easy to apply shape measures. For sufficiently smooth bodies , the Minkowski Scalars can be intuitively defined via (weighted) integrals over the volume or boundary of the body . There are three independent Minkowski scalars in 2D, and four independent in 3D.
Hadwiger’s theorem 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 :
In this sense, mathematical theory imposes that scalar Minkowski functionals are the relevant shape indices w.r.t. to physical properties represented by such additive functionals.
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).
Integrated mean curvature
(Mean width for convex bodies)
= principal curvature