Susanna Dann

The research part of this project concerns several areas of convex geometry. One part is based on the previous as well as current and ongoing work with Grigoris Paouris (Texas A&M University) and Peter Pivovarov (University of Missouri). We introduce generalizations of classical notions of affine and dual affine quermassintegrals and extend all previously known results to this new setting. It seems that further generalizations would yield promising applications, similar in spirit to those achieved in [DPP 2016]. Our results from [DPP 2016] suggest the existence of new geometric inequalities that seem to be connected with the k-plane transform. I would like to explore this direction, which will require surveying the existing literature on the subject, and is therefore a good topic for a seminar as there are many open questions related to the  k-plane transform. Another part of the project concerns questions around the Simplex Mean Width Conjecture. The last part is about smooth valuations. This is a recent and promising avenue in the theory of valuations with many open questions.

The research part of this project concerns several areas of convex geometry. One part is based on the previous as well as current and ongoing work with Grigoris Paouris (Texas A&M University) and Peter Pivovarov (University of Missouri). We introduce generalizations of classical notions of affine and dual affine quermassintegrals and extend all previously known results to this new setting. It seems that further generalizations would yield promising applications, similar in spirit to those achieved in [DPP 2016]. Our results from [DPP 2016] suggest the existence of new geometric inequalities that seem to be connected with the k-plane transform. I would like to explore this direction, which will require surveying the existing literature on the subject, and is therefore a good topic for a seminar as there are many open questions related to the  k-plane transform. Another part of the project concerns questions around the Simplex Mean Width Conjecture. The last part is about smooth valuations. This is a recent and promising avenue in the theory of valuations with many open questions.