Valuations on polytopes containing the origin in their interiors (Q1851240)

Let \({\mathcal P}_0^d\) \(({\mathcal K}^d_0)\) be the space of convex polytopes (convex bodies) in \(\mathbb{R}^d\) \((d\geq 2)\) having the origin as an interior point. Let \(\mu:{\mathcal P}^d_0 \to\mathbb{R}\) be a measurable, \(SL(d)\) invariant valuaton which is homogeneous of degree \(q,q\in \mathbb{R}\). The author obtains the following remarkable characterization. If \(q\notin \{0,d,-d\}\) then \(\mu \equiv 0\). In the other cases \(\mu\) is a constant multiple of either the Euler characteristic, the volume or the volume of the polar set. A similar result is obtained with the assumption of measurability replaced by nonnegativity. The author remarks that the result is not true with \({\mathcal P}^d_0\) replaced by \({\mathcal K}^d_0\), and that analogous classifications for \({\mathcal K}^d_0\) are unknown.