Faculty of Mathematics Professorship Geometry An Invitation to Generalized Minkowski Geometry Dissertation submitted to the Faculty of Mathematics at Technische Universität Chemnitz in accordance with the requirements for the degree doctor rerum naturalium (Dr. rer. nat.) by Dipl.-Math. Thomas Jahn Referees: Prof. Dr. Horst Martini (supervisor) Prof. Dr. Valeriu Soltan Prof. Dr. Senlin Wu submitted December 3, 2018 defended February 14, 2019 Jahn, Thomas An Invitation to Generalized Minkowski Geometry Dissertation, Faculty of Mathematics Technische Universität Chemnitz, 2019 Contents Bibliographical description 5 Acknowledgments 7 1 Introduction 9 1.1 Description of the content . 10 1.2 Vector spaces . 11 1.3 Gauges and topology . 12 1.4 Convexity and polarity . 15 1.5 Optimization theory and set-valued analysis . 18 2 Birkhoff orthogonality and best approximation 25 2.1 Dual characterizations . 27 2.2 Smoothness and rotundity . 37 2.3 Orthogonality reversion and symmetry . 42 3 Centers and radii 45 3.1 Circumradius . 46 3.2 Inradius, diameter, and minimum width . 54 3.3 Successive radii . 65 4 Intersections of translates of a convex body 73 4.1 Support and separation of b-convex sets . 77 4.2 Ball convexity and rotundity . 80 4.3 Representation of ball hulls from inside . 83 4.4 Minimal representation of ball convex bodies as ball hulls . 90 5 Isosceles orthogonality and bisectors 95 5.1 Symmetry and directional convexity of bisectors . 96 5.2 Topological properties of bisectors . 99 5.3 Characterizations of norms . 106 5.4 Voronoi diagrams, hyperboloids, and apollonoids . 107 6 Ellipsoids and the Fermat–Torricelli problem 119 6.1 Classical properties of Fermat–Torricelli loci . 120 6.2 Nonclassical properties of Fermat–Torricelli loci . 128 6.3 Boundary structure of ellipsoids . 133 3 Contents 7 Cassini sets 139 7.1 Starshapedness . 140 7.2 Connectedness . 145 8 Outlook 149 Theses 153 Bibliography 155 Curriculum vitae 169 Personal information . 169 Studies and education . 169 Published or accepted articles in journals . 169 Conference talks . 170 4 Bibliographical description Thomas Jahn An Invitation to Generalized Minkowski Geometry Dissertation, 170 pages, Technische Universität Chemnitz, Faculty of Mathematics, 2019 Report The present thesis contributes to the theory of generalized Minkowski spaces as a continuation of Minkowski geometry, i.e., the geometry of finite-dimensional normed spaces over the field of real numbers. In a generalized Minkowski space, distance and length measurement is provided by a gauge, whose definition mimics the definition of a norm but lacks the symmetry requirement. This seemingly minor change in the definition is deliberately chosen. On the one hand, many techniques from Minkowski spaces can be adapted to generalized Minkowski spaces because several phenomena in Minkowski geometry simply do not depend on the symmetry of distance measurement. On the other hand, the possible asymmetry of the distance measurement set up by gauges is nonetheless meaningful and interesting for applications, e.g., in location science. In this spirit, the presentation of this thesis is led mainly by minimization problems from convex optimization and location science which are appealing to convex geometers, too. In addition, we study metrically defined objects, which may receive a new interpretation when we measure distances asymmetrically. To this end, we use a combination of methods from convex analysis and convex geometry to relate the properties of these objects to the shape of the unit ball of the generalized Minkowski space under consideration. Keywords Apollonius circle, ball convexity, Birkhoff orthogonality, bisector, Cassini curve, ellipse, gauge, generalized Minkowski space, hyperbola, isosceles orthogonality, successive radii 5 Acknowledgments At first, I would like to express my sincere gratitude and deep appreciation to the people who shaped me as a mathematician during my research. I am indebted to my supervisor Prof. Dr. Horst Martini for his unlimited advice and support, for his trust and confidence, and for giving me the opportunity to meet lots of inspiring colleagues at wonderful conferences. Thankfully, I became a coauthor of some of these colleagues. In particular, I owe thanks to Christian Richter and Bernardo González Merino who did not only impress me with their geometric intuition but also with their kindheartedness. Furthermore, I am grateful to Prof. Dr. Gert Wanka and his former research group at Technische Universität Chemnitz (including Sebastian Banert, Radu Ioan Bot, , Ernö Robert Csetnek, Sorin- Mihai Grad, Christopher Hendrich, and Oleg Wilfer) for the scientific exchange across research groups in a friendly and supportive atmosphere. Moreover, I would like to thank Prof. Dr. Valeriu Soltan and Prof. Dr. Senlin Wu for the time and effort they invested in reviewing this thesis. 7 1 Introduction Abstraction and formalization of aspects of everyday experience is a basic ingredient for math- ematics. For instance, distance measurement is a way of quantifying the concept of the spatial gap between physical objects or the temporal gap between events. In this sense, distances have been implemented in a variety of mathematical contexts. Common features of those abstract distance notions are their nonnegativity and the validity of the triangle inequality, meaning that the distance between two points is less or equal than the accumulated distance taking a detour via a third point. The comparison of arithmetic combinations of its values is therefore a defining property of distance. Comparing distances immediately leads to optimization, i.e., asking for nearest or farthest points. This may be utilized for approximation tasks, i.e., for the process of replacing real-world data by a sufficiently simple model. In many applications, distances are symmetric in the sense that it does not matter if we measure the distance from a first to a second point or vice versa. Sometimes, however, an asymmetric formalization of distances is required by real-world applications, e.g., when we model transportation costs for uphill and downhill traveling. Norms are a common tool for defining distances between points in vector spaces which are not only symmetric but also translation-invariant and homogeneous. The shape of balls is then independent of their centers and radii. Therefore, one can show that norms correspond to centrally symmetric convex bodies in a natural way. Aiming for applications, where symmetric distance measurement is undesirable or inappropriate, a reasonable generalization of norms should still yield translation-invariant distances but allow for asymmetry. Gauges meet these requirements, yet norms have been studied more in greater detail and are considerably better understood today. Minkowski introduced gauges in his monograph Geometrie der Zahlen [167], which appeared in its first edition in 1896. In contrast to that, the term Minkowski space usually stands for a finite-dimensional normed space in modern convex geometry. Mirroring this, we shall call a finite-dimensional real vector space equipped with a gauge a generalized Minkowski space. In some situations, the translation invariance of distances allows for another viewpoint on gen- eralized Minkowski spaces. Namely, the reference to the gauge appearing in certain definitions may be replaced by a reference to its unit ball or, more generally, by a reference to a gen- eral convex body with non-empty interior. This approach is chosen, e.g., in the study of self- circumferences of convex sets in the plane [84] or in the study of optimal containment of pairs of convex sets [35, 37, 142]. Explicit reference to a gauge appears, e.g., in the study of bisec- tors [144] or in certain convex optimization problems [195]. Topics such as the inscribability of simplices in smooth hypersurfaces [94] may also be seen in this context. The present thesis is a contribution to the theory of generalized Minkowski spaces. In the se- 9 1 Introduction quel, we discuss generalizations of a variety of metrically defined objects from Minkowski spaces and look for replacements for the corresponding theorems. As it turns out, the seemingly minor change in the definition of a norm amounts to a much richer theory in some places in the sense that we will encounter phenomena which are exclusive to norms, and others which are exclu- sive to non-norms. Continuing the interaction between analysis and geometry, the shape of a distinguished convex body, namely the unit ball, will play a crucial role for analytic properties of the gauge such as differentiability. In this context, rotundity, which is one way of formalizing the roundness of convex bodies, will be a recurring topic. 1.1 Description of the content The presentation of thesis at hand follows the one given in our articles. For the sake of con- sistency, some notational changes and rearrangements of results had to be performed. In the remainder of the present chapter, we provide basic definitions and results from convex geometry and convex analysis. This way, we fix the language of generalized Minkowski spaces analogously to the one of normed spaces. Chapter 2 deals with Birkhoff orthogonality. This binary relation is a generalization of usual Euclidean orthogonality. It is closely connected to the approximate solutions of a variant of the best approximation problem. Based on the results of [121, Section 3], we give dual descriptions of "-Birkhoff orthogonality, show how smoothness and rotundity of the gauge impacts 0-Birkhoff orthogonality,