Hadwiger's theorem

Remarks

Conversely, each $V_j$ is a continuous valuation invariant under rigid motions, so these valuations form a real vector space of dimension $n + 1$ with basis $V_0, \dots, V_n$. A short proof by Klain rests on a lemma about simple valuations. The lemma says: a continuous valuation invariant under rigid motions that vanishes on all sets of dimension below $n$ is a multiple of volume. Induction on $n$ then handles the restriction of $\varphi$ to lower-dimensional sets.

Example: surface area in $\R^n$ is a continuous valuation invariant under rigid motions, and the theorem identifies it as $2 V_{n-1}$.

Invariance under rotations is needed. In $\R^2$ the width in the direction of the first axis, $\varphi(K) = \max_{x \in K} x_1 - \min_{x \in K} x_1$, is a continuous valuation invariant under translations. It gives $1$ to the segment $[0,1] \times \set{0}$ and $0$ to the congruent segment $\set{0} \times [0, 1]$, so it is not a combination of intrinsic volumes.

Going deeper

For a student

The theorem is the foundation of integral geometry in Euclidean space. It gives quick proofs of the principal kinematic formula and of Crofton-type formulas: one checks that both sides are such valuations and compares them on balls. It also explains why intrinsic volumes appear in probability, for example in the expected Euler characteristic of random sets. Its classification has been extended by Alesker and others to valuations invariant under smaller groups.