\(\mathrm{SL}(n)\) invariant valuations on polytopes (Q527435)

A \textit{valuation} \(\Phi\) is a real function on a family of sets in \(\mathbb R^{n}\) that satisfies the relation \[ \Phi (K)+\Phi (L)=\Phi (K\cup L)+\Phi (K\cap L) \] whenever \(K, L, K\cup L, K\cap L\) are sets of this family. Important classical contributions on valuations are due to Dehn, Blaschke and Hadwiger (Hadwiger's characterization theorem). Let \({\mathcal K}^{n}\) be the family of all convex bodies in \(\mathbb R^{n}\). The authors prove the following basic results: Theorem 2: \(\Phi : {\mathcal K}^{n}\to\mathbb R\) is an upper semicontinuous translation and \(SL(n)\) invariant valuation if and only if there are constants \(c_0, c_1\in\mathbb R\) and \(c_2\geq 0\) such that \(\Phi = c_0 V_0 + c_1 V_n + c_2 \Omega\) where \(V_0\) is the Euler characteristic, \(V_n\) the volume and \(\Omega\) the (generalized) affine surface area. Let \({\mathcal K}_0^{n}\) be the family of all convex bodies in \(\mathbb R^{n}\) with the origin as an interior point. Theorem 3: \(\Phi : {\mathcal K}_0^{n}\to\mathbb R\) is an upper semicontinuous \(GL(n)\) invariant valuation if and only if there are constants \(c_0\in\mathbb R, c_1>0\) such that \(\Phi = c_0 V_0+c_1\Omega_c\). Here \(\Omega_c\) is the centro-affine surface area. In Theorem 4, the authors characterize the upper semicontinuous \(SL(n)\) invariant valuations which are homogeneous of degree \(q\in \mathbb R\), and in Theorem 5 they characterize the \(SL(n)\) invariant valuations which vanish on the family of polytopes. The proofs of these deep results are complicated and make use of recent results on affine and \(L_p\)-affine surface area. Another tool is a new shaping process for convex bodies.