Seminar for Dynamic Systems

On Friday, March 14, 2025, 16:00 hours, lecture room 104, Steffen Winter (Karlsruhe University) will hold lecture titled:

On support measures, Minkowski content and complex dimensions.

 *Abstract:* 

For any given closed set inR^d, the general Steiner formula of Hug, Last and Weil describes its tube volume and the support measures arising from this formula encode its geometric properties. However, it is not easy to extract fractal properties of a set directly from these measures. We introduce some new geometric functionals for compact subsets of R^d which we call basic functionals and support functionals, which are tightly related to the support measures. We connect appropriate scaling exponents of these functionals to the (upper) Minkowski dimension of the set and explain how its Minkowski content may be represented in terms of these functionals. Our original motivation was to elucidate the geometric meaning of the coefficients in fractal tube formulas that arise in the theory of complex dimensions by Lapidus, Zubrinic and Radunovic. We explain the relations and also comment on the connections to fractal curvatures. Based on joint work with Goran Radunovic.