Numerical evaluation of multi-loop integrals

We present updates on the development of pySecDec, a toolbox to numerically evaluate parameter integrals in the context of dimensional regularization. We discuss difficulties with loop integrals in the special kinematic condition where the squared momentum of a leg is equal to the squared mass of a propagator. We further discuss some features of a Quasi Monte Carlo (QMC) integrator that can optionally run on Graphics Processing Units (GPUs).Comment: 10 pages, 5 figures, contribution to the proceedings of Loops and Legs 2018, St. Goar, German

Successive Radii and Ball Operators in Generalized Minkowski Spaces

We investigate elementary properties of successive radii in generalized Minkowski spaces (that is, with respect to gauges), i.e., we measure the "size" of a given convex set in a finite-dimensional real vector space with respect to another convex set. This is done via formulating some kind of minimal containment problems, where intersections or Minkowski sums of the latter set and affine flats of a certain dimension are incorporated. Since this is strongly related to minimax location problems and to the notions of diametrical completeness and constant width, we also have a look at ball intersections and ball hulls.Comment: submitted to "Advances of Geometry