<> "The repository administrator has not yet configured an RDF license."^^ . <> . . "The horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces"^^ . "This work examines the horofunction compactification of finite-dimensional normed vector spaces\r\nwith applications to the theory of symmetric spaces and toric varieties.\r\nFor any proper metric space X the horofunction compactification can be defined as the closure of\r\nan embedding of the space into the space of continuous real valued functions vanishing at a given\r\nbasepoint. A point in the boundary is called a horofunction. This characterization though lacks\r\nan explicit characterization of the boundary points. The first part of this thesis is concerned with\r\nsuch an explicit description of the horofunctions in the setting of finite-dimensional normed vector\r\nspaces. Here the compactification strongly depends on the shape of the unit and the dual unit ball\r\nof the norm. We restrict ourselves to cases where at least one of the following holds true:\r\nI) The unit and the dual unit ball are polyhedral.\r\nII) The unit and the dual unit ball have smooth boundaries.\r\nIII) The metric space X is two-dimensional.\r\nBased on a result of Walsh we provide a criterion for the convergence of sequences in\r\nthe horofunction compactification in these cases to determine the topology. Additionally we show\r\nthat then the compactification is homeomorphic to the dual unit ball. Later we give an explicit\r\nexample, where our criterion for convergence fails in the general case and make a conjecture\r\nabout the rate of convergence of some spacial sets in the boundary of the dual unit ball. Assuming\r\nthe conjecture holds, we generalize the convergence criterion to any norm with the property that all\r\nhorofunctions in the boundary are limits of almost-geodesics (so-called Busemann points). This\r\npart of the thesis ends with a construction of how to extend our previous results to a new class of\r\nnorms using Minkowski sums:\r\nIV) The dual unit ball is the Minkowski sum of a polyhedral and a smooth dual unit ball.\r\nThe second part of the thesis applies the results of part one to two different settings: first to sym-\r\nmetric spaces of non-compact type and then to projective toric varieties. For a symmetric space\r\nX = G/K of non-compact type with a G-invariant Finsler metric we prove that the horofunction\r\ncompactification of X is determined by the horofunction compactification of a maximal flat in X.\r\nWith this result we show how to realize any Satake or Martin compactification of X as an appropri-\r\nate horofunction compactification. Finally, as an application to projective toric varieties, we give a\r\ngeometric 1-1 correspondence between projective toric varieties of dimension n and horofunction\r\ncompactifications of R^n with respect to rational polyhedral norms."^^ . "2021" . . . . . . . "Anna-Sofie"^^ . "Schilling"^^ . "Anna-Sofie Schilling"^^ . . . . . . "The horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces (PDF)"^^ . . . "Dissertation_Anna_Schilling_PDF_A.pdf"^^ . . . "The horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces (Other)"^^ . . . . . . "lightbox.jpg"^^ . . . "The horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces (Other)"^^ . . . . . . "preview.jpg"^^ . . . "The horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces (Other)"^^ . . . . . . "medium.jpg"^^ . . . "The horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces (Other)"^^ . . . . . . "small.jpg"^^ . . . "The horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces (Other)"^^ . . . . . . "indexcodes.txt"^^ . . "HTML Summary of #30380 \n\nThe horofunction compactification of finite-dimensional normed vector spaces and of symmetric spaces\n\n" . "text/html" . . . "510 Mathematik"@de . "510 Mathematics"@en . .