On simple frame valuations in many dimensions (Q1374363)

Let \(P\) be a polytope in Euclidean space \(\mathbb{R}^{n}\). A flag \(f=(f_{0},f_{1},\dots,f_{n-1})\) is said to be produced by \(P\) if each \(k\)-dimensional flat \(f_{k}\), \(k=0,1,\dots,n-1,\) contains a \(k\) -dimensional face of \(P\). To a flag produced by \(P\) corresponds an orthogonal frame \(\bar{f}=(f_{0},\omega _{1},\dots,\omega _{n-1}),\) where the unit vector \(\omega _{k},\) \(k=0,1,\dots,n-1\), is an outward normal to \(P\) belonging to the orthogonal complement of \( f_{k}\). For any real function \(F\) defined on the set of frames the frame valuation with primary function \(F\) is defined by \(\Psi _{F}(P):=\sum F(\bar{f})\) where the sum ranges over the frames associated to the flags produced by \(P\). The author proves that all simple valuations defined on polytopes can be represented as frame valuations. This result is based on the representation of the indicator function of \(P\) as finite linear combination of indicator functions of the Schläfli simplices associated to the flags produced by \(P\). The author also obtains a sufficient condition to guarantee that a frame valuation generates a signed measure. This condition turns out to be necessary for the class of the so-called Ostrogradski valuations.