The multiplicative structure on continuous polynomial valuations
Let \({\mathcal K}(V)\) be the class of all convex compact sets of a real linear vector space \(V\) with \(\dim V=n\). A function \(\phi: {\mathcal K}(V)\to\mathbb{C}\) is called a continuous valuation if it is continuous with respect to the Hausdorff metric of \({\mathcal K}(V)\) and satisfies \[ \phi(K_1\cup K_2)=\phi(K_1)+\phi(K_2)-\phi(K_1\cap K_2) \] provided \(K_1\cup K_2\) is convex. It is said to be polynomial of degree \(d\) if for all \(K\in {\mathcal K}(V)\) the function \(x\mapsto \phi(K+x)\) is a polynomial on \(V\) of degree at most \(d\). Using representations of the linear group \(GL(V)\) acting transitively on \({\mathcal K}(V)\), the author introduces a subclass of polynomial valuations which are \(GL(V)\)-smooth. This subclass has a rich algebraic structure (filtration, multiplication, etc.) turning it in a commutative associated filtered algebra. As an application of the author's study of the algebraic structure for spaces of valuation the following generalization of well-known results of Hadwiger and McMullen is proved (Theorem 0.9). Let \(G\) be a compact subgroup of \(GL(V)\). Let \(\text{Val}^G(V)\) and \(\text{Val}^G_i(V)\) be the spaces of \(G\)-invariant valuations and \(i\)-homogeneous valuations, respectively. Theorem. Assume that \(G\) acts transitively on the projective space \(\mathbb{P}(V)\). Then: (a) \(\text{Val}^G(V)=\bigoplus^n_{i=0}\text{Val}^G_i(V)\) and \(h_i:=\dim\text{Val}^G_i <\infty\). (b) \(\text{Val}^G(V)\) is a graded algebra satisfying Poincaré duality; in particular, \(h_i=h_{n-i}\). (c) \(\text{Val}^G_i(V)\) is spanned by the intrinsic volume \(\text{vol}^i\), \(i=1\) or \(n-1\), where the intrinsic volumes are taken with respect to a \(G\)-invariant Euclidean metric \((\rho_0, h=h_{n-1}=1)\). (d) If, in addition, \(-\text{Id}\in G\), then the Lefschetz inequalities \(h_i\leq h_{i+1}\) hold for \(i<n/2\).
commutative associative filtered algebra
scientific article; zbMATH DE number 2121749