A simplified elementary proof of Hadwiger's volume theorem (Q1827418)

If we ``fatten up'' a \(d\)-dimensional convex body \(K\) by putting a coat of paint with constant thickness \(A\) onto it (that is, replace \(K\) by the Minkowski sum \(K+ \lambda B^d\)) we increase its volume. A classic result of Steiner shows that the new volume has a surprisingly simple form; it is a polynomial of degree \(d\) in \(X\). The coefficient of degree \((d-i)\) involves a quantity \(V_i(K)\), called the \(i\)th intrinsic volume of \(K\). For \(i= 0\), \(1\), \(d-1\), \(d\), the intrinsic volumes are the Euler characteristic, mean width, surface measure and volume respectively. The intrinsic volumes are all invariant under isometries, valuations (that is, \(V_i(K)+ V_i(L)= V_i (K\cap L)+ V_i(K\cup L)\)), and continuous (when \(K_n\to K\) in the Hausdorff metric, then \(V_i(K_n)\to V_i(K)\)). Hadwiger's intrinsic volume theorem states that the intrinsic volumes \((V_i:0\leq i\leq d)\) form a basis for the continuous isometry-invariant valuations on the set of compact convex sets in \(\mathbb{R}^d\). A valuation \(V\) is called simple if \(V(K)= 0\) when \(K\) does not have full dimension. The only simple intrinsic volume is \(V_d\), the \(d\)-dimensional volume; and the intrinsic volume theorem implies trivially (and may easily be proved from) Hadwiger's volume theorem, which states that (up to scalar multiplication) \(V_d\) is the only simple continuous isometry-invariant valuation on the set of compact convex sets in \(\mathbb{R}^d\). For a long time only Hadwiger's rather daunting 1957 proof was available for these results; more recently, Klain gave a simpler proof, but one which used rather advanced methods involving spherical harmonics. The Volume Theorem is proved here, by an ingenious, elementary and comparatively straightforward method. The proof begins with a sequence of lemmas. Some of these show that fairly general classes of polytopes may be dissected into direct sums and other special polytopes; others relate valuations on special classes of polytopes to those on more general classes. Lemma 3.4, which states that a simplex may be dissected into two homothetical simplexes and various direct sums of faces, perhaps requires a little comment. The notation of formula (6) on page 117 is a little casual, and suggests that the two simplexes are dilatations rather than homothets of the original simplex -- but once this is straightened out the lemma is straightforward. The sketch given here (which might have clarified the paper) illustrates the simplest nontrivial case. \[ \begin{tikzpicture}[line width=1pt] \draw (3,0) -- (0,0) -- (0,3) -- cycle; \draw (2,0) -- (0,2); \draw (2,0) -- (2,1); \end{tikzpicture} \] Finally, all these elements are put together to prove that a continuous isometry-invariant simple valuation on compact convex sets that vanishes on the unit cube is identically zero, from which Hadwiger's theorems follow directly. To summarize, this is an elegant and accessible simplification of a very important proof. I imagine that it will be presented in many textbooks and graduate seminars in years to come.