VECTOR LATTICES, POLYHEDRAL GEOMETRY, AND VALUATIONS

The present thesis explores the connections between Hadwiger’s Characterization Theorem for valuations, vector lattices, and MV-algebras. The study of valuations can be seen as a precursor to the measure theory of modern probability. For this reason, valuations are one of the most important topics of geometric probability. One of the topics that turns out to be central is the study of measures on polyconvex sets (i.e., finite unions of compact convex sets) in Euclidean spaces of arbitrary finite dimension, that are invariant under the group of Euclidean motions. Hadwiger’s Characterization Theorem states that the linear space of such invariant measures is of dimension n + 1, if the ambient has dimension n. Moreover, its proof shows that the Euler-Poincaré characteristic is a basic invariant measure. The Euler-Poincaré characteristic, indeed, is the unique such measure that assigns value one to each compact and convex set. The main topic of the first part of our work is the characterization of the Euler-Poincaé characteristic as a valuation on finitely presented unital vector lattices. By the Baker-Baynon duality, we represent each finitely presented unital vector lattice as the lattice of continuous and piecewise linear real-valued functions on a suitable polyhedron in the Euclidean space. Then we define vl-Schauder hats, that are special elements of the vector lattice with a “pyramidal shape”, and that can be used to generate the vector lattice, via addition and products by real scalars. On the positive cone of every finitely presented vector lattice V we define a pc-valuation as a valuation (in the usual classical sense) that is insensitive to addition. The (Euler-Poincaré) number χ(f) of any function f in the positive cone of V is next defined as the Euler-Poincaré characteristic of the support of f. Pc-valuations uniquely extend to a suitable kind of valuations over V, called vl-valuations. We then prove a Hadwiger-like theorem, to the effect that our χ is the only vl-valuation assigning 1 to each vl-Shauder hat of V. In the second part of the thesis we explore two different ways to associate continuous and piecewise linear functions (and hence elements of vector lattices) to geometric objects. First we use the notion of support function to establish a correspondence between a suitable subset of the free vector lattice on n generators and the set of polytopes of the n-dimensional euclidean space. This special set of algebraic objects generates the whole free vector lattice via finite meets. We call it the set of support elements. Then we consider valuations on the free vector lattice that are also additive on the set of support elements. By the Volland-Groemer Extension Theorem, we prove that such valuations are in a one-to-one correspondence with the valuations on the lattice of polyconvex sets that are additive on the subset of convex objects. Next we proceed in a similar way, using gauge functions. In this case, the first correspondence that we prove is between the positive cone of the vector lattice of continuous and positively homogeneous real-valued functions of the n-dimensional euclidean space and a lattice, equipped with appropriate vector space operations, of a new kind of geometric objects. We call these sets star-shaped objects. Then we define a geometric notion of good sequence (a tool introduced by D.Mundici in the study of MV-algebras), and we prove a representation theorem for star-shaped objects, in terms of good sequences. Imposing a polyhedral condition on our star-shaped objects, we obtain a correspondence between them and the elements of the positive cone of the free vector lattice on n generators. Finally, we specialize the result obtained for good sequences to these polyhedral star-shaped objects.