Lines Tangent to 2n-2 spheres in R^n

We show that there are 3 \cdot 2^(n-1) complex common tangent lines to 2n-2 general spheres in R^n and that there is a choice of spheres with all common tangents real.Comment: Minor revisions. Trans. AMer. Math. Soc., to appear. 15 pages, 3 .eps figures; also a web page with computer code verifying the computations in the paper and with additional picture

Transversals to line segments in three-dimensional space

http://www.springerlink.com/We completely describe the structure of the connected components of transversals to a collection of $n$ line segments in $\mathbb{R}^3$. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the non-generic cases and show that $n\geq 3$ arbitrary line segments in $\mathbb{R}^3$ admit at most $n$ connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of $n$ on the number of geometric permutations of line segments in $\mathbb{R}^3$

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