Seminar

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First, we are going to discuss the following generalization of the classical boxing inequality:

Given a n-dimensional manifold M in a finite or infinite-dimensional Banach space B, and a real number m in (0,n], there exists a (n+1)-dimensional pseudomanifold W in B such that: (1) M is the boundary of W, and (2) the m-dimensional

Hausdorff content of W does not exceed c(m) multiplied by the m-dimensional Hausdorff content of M.

(Recall that the m-dimensional Hausdorff content of X is defined as follows: For each covering of X by metric balls consider the sum of the m-th powers of the radii of these balls, and then take the infimum of these sums over all coverings of X.)

Then we will discuss further generalizations of this result, its connections to systolic geometry, and with Urysohn width-volume inequalities. (The informal meaning of the first such inequality that was proven by L. Guth is that each Riemannian manifold with a small volume must be ``close" to a polyhedron of a lower dimension. )

Finally, we will present two new width - volume inequalities and discuss their implications for systolic geometry.

Joint work with Sergey Avvakumov.