An Invitation to Generalized Minkowski Geometry

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

Renewal theorems for a class of processes with dependent interarrival times and applications in geometry

Renewal theorems are developed for point processes with interarrival times $W_n=\xi(X_{n+1}X_n\cdots)$, where $(X_n)_{n\in\mathbb Z}$ is a stochastic process with finite state space $\Sigma$ and $\xi\colon\Sigma_A\to\mathbb R$ is a H\"older continuous function on a subset $\Sigma_A\subset\Sigma^{\mathbb N}$. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry; for instance, results on the Minkowski measurability of self-conformal sets are deduced. Indeed, these geometric problems motivated the development of the renewal theorems.Comment: 2 figure