Integral Geometry

In n-dimensional Euclidean space, the Minkowski functionals can be defined by the Steiner formula [1]. Given a convex body K, i.e., a compact, convex subset of \mathbb{R}^n, the parallel body K_{\epsilon} at a distance \epsilon \geq 0 is the set of all points x for which there is a point y in K such that the distance between x and y is smaller or equal to \epsilon. The volume W_0 of this parallel body K_{\epsilon} can be expressed by a polynomial of \epsilon

W_0(K_{\epsilon}) = W_0(K) + \frac{1}{n}\sum_{\nu=1}^n\epsilon^{\nu}\cdot\binom{n}{\nu}\cdot W_{\nu}(K),

where W_0(K)=V_n(K) and the coefficients W_1(K),\dots,W_n(K) depend on K but not on \epsilon. The so-called Minkowski functionals W_1(\cdot),\dots,W_n(\cdot) can be generalized to finite unions of convex bodies, so that

W_{\nu}(K\cup L) = W_{\nu}(K) + W_{\nu}(L) - W_{\nu}(K\cap L).

This is the additivity property [1]. By this choice of normalization, the zeroth order Minkowski functional of an n-dimensional unit ball B^n=\{x\in\mathbb{R}^n\mid\|x\|\leq 1\} is equal to its volume W_0(B^n)=:\kappa_n, and the higher-order Minkowski functionals are equal to its surface area W_{\nu}(B^n)=n\kappa_n (n\geq \nu\geq 1). In the mathematical literature the intrinsic volumes V_{\nu} are more commonly used than the Minkowski functionals, but only differ by a proportionality constant and the order of the indices

V_{\nu}(K) :=\frac{1}{n\cdot\kappa_{n-\nu}}\cdot{{n}\choose{\nu}}\cdot W_{n-\nu}(K)\quad\mathrm{for\ }\nu\leq n-1.

The Minkowski tensors of a convex body K in the d-dimensional Euclidean space can be defined using the so-called support measures \Lambda_{\nu}(K;\cdot) [1], which can in turn be defined by a local Steiner formula. Their total mass yields the intrinsic volumes, that is V_{\nu}=\Lambda_{\nu}(K;\mathbb{R}^n\times\mathbb{S}^{n-1}). Like the scalar functionals, the tensors can be generalized to non-convex bodies by using their additivity [1]. For consistency with the Minkowski functionals, we use for \nu\geq 1 a different normalization \Omega_{\nu}(K;\cdot):=\frac{n\cdot\kappa_{n-\nu}}{\binom{n}{\nu}}\Lambda_{n-\nu}(K;\cdot). The Minkowski tensors are then defined as [1, 2]

W_{0}^{r,0}(K):= \int_{K}\mathbf{x}^r\text{d}^n\,\mathbf{x},

W_{\nu}^{r,s}(K):= \int_{\mathbb{R}^n\times\mathbb{S}^{n-1}}\mathbf{x}^r\mathbf{u}^s\,\Omega_{\nu}(K;\text{n}(\mathbf{x},\mathbf{u})) \quad\mathrm{for\ }n\geq \nu \geq 1,

where \mathbf{x}^r or \mathbf{u}^s are symmetric tensor products, and \mathbf{x}^r\mathbf{u}^s is the symmetric tensor product of the tensors \mathbf{x}^r and \mathbf{u}^s.

(The presentation follows reference [3].)

[1] R. Schneider and W. Weil, Stochastic and integral geometry, Springer Science & Business Media, 2008.
[2] P. McMullen, “Isometry covariant valuations on convex bodies,” Supplemento ai rendiconti circ mat palermo, vol. 50, p. 259–271, 1997.
[3] [doi] M. A. Klatt, G. Last, K. Mecke, C. Redenbach, F. M. Schaller, and G. E. Schröder-Turk, “Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors,” in Tensor Valuations and Their Applications in Stochastic Geometry and Imaging, Springer, Cham, 2017, p. 385–421.