The Horofunction Compactification of Finite-Dimensional Normed Spaces and of Symmetric Spaces.

This work examines the horofunction compactification of finite-dimensional normed vector spaces with applications to the theory of symmetric spaces and toric varieties.

For any proper metric space \(X\) the horofunction compactification can be defined as the closure of an embedding of the space into the space of continuous real valued functions vanishing at a given basepoint. A point in the boundary is called a horofunction. This characterization though lacks an explicit characterization of the boundary points. The first part of this thesis is concerned with such an explicit description of the horofunctions in the setting of finite-dimensional normed vector spaces. Here the compactification strongly depends on the shape of the unit and the dual unit ball of the norm. We restrict ourselves to cases where at least one of the following holds true:

I) The unit and the dual unit ball are polyhedral.

II) The unit and the dual unit ball have smooth boundaries.

III) The metric space \(X\) is two-dimensional.

Based on a result of Walsh we provide a criterion for the convergence of sequences in the horofunction compactification in these cases to determine the topology. Additionally we show that then the compactification is homeomorphic to the dual unit ball. Later we give an explicit example, where our criterion for convergence fails in the general case and make a conjecture about the rate of convergence of some spacial sets in the boundary of the dual unit ball. Assuming the conjecture holds, we generalize the convergence criterion to any norm with the property that all horofunctions in the boundary are limits of almost-geodesics (so-called Busemann points). This part of the thesis ends with a construction of how to extend our previous results to a new class of norms using Minkowski sums:

IV) The dual unit ball is the Minkowski sum of a polyhedral and a smooth dual unit ball.

The second part of the thesis applies the results of part one to two different settings: first to sym- metric spaces of non-compact type and then to projective toric varieties. For a symmetric space \(X = G/K\) of non-compact type with a \(G\)-invariant Finsler metric we prove that the horofunction compactification of \(X\) is determined by the horofunction compactification of a maximal flat in \(X\). With this result we show how to realize any Satake or Martin compactification of \(X\) as an appropri- ate horofunction compactification. Finally, as an application to projective toric varieties, we give a geometric 1-1 correspondence between projective toric varieties of dimension n and horofunction compactifications of \(\mathbb{R}^n\) with respect to rational polyhedral norms.