Minimal distances and optimal paths

Přednáška prof. Guntera Weisse
působícího na Technických univerzitách v Drážďanech a ve Vídni.

Anotace přednášky: We are used to a line segment being the shortest connection between two points. But this seemingly obvious fact is based on assumptions. What does it mean “to be minimal distant” in space and in what kind of space or in time? Here geometry, physics, mathematics (optimisation), and even biology has an impact.
Assuming the place of action being a Euclidean plane or space, one can consider minimal distances, where the path fulfils conditions. Here using reflections, i.e. mappings which are distance preserving, help to find the solution. This principle is applicable e.g. to billiards. The shortest connection of two points on a surface is a problem of Differential Geometry, the result is called a “geodesic arc”. In Euclidean planes geodesic arcs are just line segments and one might consider the “geodesic polygon” between two points of a polyhedron. Packing e.g. a box with a rope induces the concept “closed geodesic polygon” on a polyhedron and challenges to deeper investigation.
Finally, the shortest connection between a set of points in space will shortly be considered, too. It connects elementary geometry to the well-known concepts “Steiner’s minimal tree” and the “traveling salesman problem”.