A short proof of Hadwiger's characterization theorem,
Mathematika, 42 1995, 329-339.
One of the most beautiful and important results in geometric
convexity is Hadwiger's characterization theorem for the
quermassintegrals. Hadwiger's theorem classifies all continuous
rigid motion invariant valuations on convex bodies as consisting of the
linear span of the quermassintegrals (or, equivalently, of the intrinsic
volumes). Hadwiger's characterization leads to effortless
proofs of numerous results in integral geometry, including various
kinematic formulas and mean projection formulas
for convex bodies. Hadwiger's result also provides a
connection between rigid motion invariant set functions and symmetric
The purpose of this paper is to present a new and shorter proof of
Hadwiger's characterization theorem, digestible within a few minutes.
En route to this result is a more general characterization of volume in
Euclidean space, as the unique continuous translation invariant simple
even valuation on compact convex sets.
A pdf file for this paper is available
Characterization of even valuations is treated in the
Even valuations on convex bodies.
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