Background – Integral geometry – Minkowski Functionals

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 $\mathbf{u}^s$ .

[1] Schneider, R., Weil, W.: Stochastic and Integral Geometry (Probability and Its Applications). Springer, Berlin (2008)

[2] McMullen, P.: Isometry covariant valuations on convex bodies. Rend. Circ. Mat. Palermo (2) Suppl. 50, 259 (1997)

[3] Klatt et al.: Cell Shape Analysis of Random Tessellations Based on Minkowski Tensors. In Tensor Valuations and Their Applications in Stochastic Geometry and Imaging (eds. Vedel Jensen, E. B. & Kiderlen, M.) 2177, 385-421 (Springer International Publishing, 2017)