An application of valuation theory to two problems in discrete geometry (Q1078811)

Let \({\mathcal P}^ d\) be the class of all convex polytopes in the euclidian d-space \(E^ d\). By definition, a valuation is a functional \(\phi\) on \({\mathcal P}^ d\) satisfying \(\phi (P\cup Q)+\phi (P\cap Q)=\phi (P)+\phi (Q)\) whenever P,Q,P\(\cup Q\in {\mathcal P}^ d\). It is called simple if \(\phi (P)=0\) for all P with \(int (P)=\emptyset.\) The authors outline a new, short and straight forward construction expressing every valuation by simple one. Using this technique it is shown that for a lattice zonotope Z (with respect to \({\mathbb{Z}}^ d\subset E^ d)\) the Wills-conjecture \(\ll G(Z)\leq W(Z)\gg\) holds (where G is the lattice point enumerator and W is the Wills' functional constructed on ''volume'' functional V). As a second application it is shown that the L. Fejes-Tóth sausage- conjecture \(V(S_ k+B^ d)\leq V(C_ k+B^ d)\) is true if \(C_ k\) is a zonotope and \(d\geq 5\) (here \(B^ d\subset E^ d\) is the unit ball, \(C_ k\) is the convex hull of the centers of k non-overlapping translations of \(B^ d\) and \(S_ k\) is a line segment of length 2(k- 1)).