Minkowski Functionals

Let M be a smooth closed bounded convex set in ℝⁿ. For r ≥ 0 let dil(M, r) be the set of all points within distance r of some point in M. (dil(M, r) is M dilated by r.) μ(dil(M, r)) = ∑_0≤i≤nf_irⁱ where
f₀ = μ(M), the volume of M,
f₁ = ∫( ∑_0≤j<n1/R_j), (n−1) (total mean curvature of surface),
...
f_j = ∫Sym_j(1/R₁, 1/R₂...), is the jth elementary symmetric polynomial of the principle curvatures of the surface of M.

The integrals are over the surface of M. μ is a measure.

The f_i are called “Minkowski Functionals”. The formula works for non smooth convex bodies. The equation for μ(dil(M, r)) can be taken as a definition for the f_i’s.

The introduction to this paper gives a good intuitive introduction to Minkowski functionals.

Consider n = 3. Relax the smooth requirement on M. Consider a closed region P on the surface of M. Consider the set S of oriented planes thru some point of P but which do not cut M. For smooth points of P there will be just one such plane and that will be the tangent plane. For non-smooth points there will be many planes. If M is flat (not strictly convex) one plane of S may intersect many points of P. The set S will belong to the Grassmann manifold G_3,2 and have a measure there. If P is the entire surface of M then that measure will be f₁.

Hadwiger’s theorem shows how Minkowski’s functionals are fundamental and unique.

It is instructive to see the functionals for the unit cube. The unit cube dilated by ε > 0 is the disjoint union of

0: the cube proper, (v=1)
1: 6 square slabs adjacent to the faces of the cube, each 1×1×ε, (v=6ε)
2: 12 truncated quarter cylinders, radius ε, touching the edges of the cube and length 1, (v=3πε²)
3: 8 eights of spheres, radius ε at the corners of the cube. (v=4π/3ε³)

Together these exhaust the dilated cube and the total volume is:
1 + 6ε + 3πε² + 4π/3ε³. The coefficients of this polynomial in ε are the functionals for the unit cube.