Time：14:00-15:45, Wednesday, November 20 2024
Venue: E10-212
Speaker：Ioann Vasilyev, Steklov Mathematical Institute
Biography: Ioann Vasilyev is a researcher at St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences. His research interests include Geometric measure theory and Harmonic and functional analysis.
Title: Convexity and minimality of flats in finite dimensional Banach spaces
Abstract: My talk will be devoted to the following question: ‘Do flats minimize the Hausdorff measure in finite dimensional Banach spaces?’ It turns out that the property mentioned in this question is strongly connected to different variants of convexity of Busemann–Hausdorff area densities. The answer to this question is not known in general and the only two known results are those concerning flats of codimension one (H. Busemann,1949) and those of dimension two (D. Burago and S. Ivanov, 2012). In the first part of my talk I will define the main objects and recall the nowadays classical results by H. Busemann, D. Burago and S. Ivanov and by others. In the second part of my talk I will show how prove the minimality in the codimension two case in complex finite dimensional Banach spaces. Also, I will show that a certain convexity property of Busemann–Hausdorff area densities fails in some spaces.
