Colloquium

January 15, 2009

4:00 PM, BAC 102

Speaker: Mikhail Ostrovskii, St. John's University

Title:  Minimal-volume projections of cubes, Sobolev spaces on graphs, and sufficient enlargements for normed spaces

At the beginning of the talk we consider the following problem: Let $K^m$ be an $m$-dimensional cube and $L$ be a linear subspace in $\mathbb{R}^m$. We consider all linear (not necessarily orthogonal) projections $P$ of $\mathbb{R}^m$ onto $L$. By compactness, there are projections which minimize the volume of $P(K^m)$. We call them {\it minimal-volume projections}. The problem is to describe the possible shapes of $P(K^m)$, where $P$ is a minimal-volume projection. It turns out that the possible shapes of $P(K^m)$ can be described in terms of totally unimodular matrices. One of the fruitful approaches to the study of the obtained class of convex bodies is by using the observation that in many cases the dual bodies are unit balls of Sobolev spaces on graphs. This observation allows to prove results about minimal-volume projections by analogy with known results about classical Sobolev spaces. The results of minimal-volume projections of cubes can be used to describe minimal-volume sufficient enlargements for normed linear spaces. Recall the definition: A symmetric convex body $A$ in a finite-dimensional normed space $X$ is called a {\it sufficient enlargement} of $B(X)$ (the unit ball of $X$) if for arbitrary isometric embedding of $X$ into any Banach space $Y$ there exists a linear projection $P:Y\to X$ such that $P(Y)=X$ and $P(B(Y))\subset A$.