In -dimensional Euclidean space, the Minkowski functionals can be defined by the Steiner formula . Given a convex body , i.e., a compact, convex subset of , the parallel body at a distance is the set of all points for which there is a point in such that the distance between and is smaller or equal to . The volume of this parallel body can be expressed by a polynomial of
where and the coefficients depend on but not on . The so-called Minkowski functionals can be generalized to finite unions of convex bodies, so that
This is the additivity property . By this choice of normalization, the zeroth order Minkowski functional of an -dimensional unit ball is equal to its volume , and the higher-order Minkowski functionals are equal to its surface area (). In the mathematical literature the intrinsic volumes are more commonly used than the Minkowski functionals, but only differ by a proportionality constant and the order of the indices
The Minkowski tensors of a convex body in the -dimensional Euclidean space can be defined using the so-called support measures , see , which can in turn be defined by a local Steiner formula. Their total mass yields the intrinsic volumes, that is . Like the scalar functionals, the tensors can be generalized to non-convex bodies by using their additivity, see . For consistency with the Minkowski functionals, we use for a different normalization . The Minkowski tensors are then defined as [1, 2]
where or are symmetric tensor products, and is the symmetric tensor product of the tensors and .
(The presentation follows reference .)