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Presented By: Department of Mathematics

Student Probability

Remark on approximating convex body by polytopes with few facets

Let K be an n-dimensional convex body such that the unit Euclidean ball, $B_2^n$, is the maximal volume ellipsoid lies inside K. (John's position) Then,
there exists a polytope P with $O(n^2)$ facets satisfying $K\subset P \subset nK$. In the case when $K$ is symmetric, then, we can reduce the coefficient n to $\sqrt{n}$.

Recently, Barvinok generalize this method to get a sharp result: For any symmetric convex body K, there exists a polytope with $m$ facets such that
$K\subset P \subset C\sqrt{\frac{n}{\log(m/n)}}K$.

In this talk, we will present the following result in the non-symmetric cases:
Let $n>R_n\ge 1$ be a sequence such that $\lim_{n\rightarrow \infty} \frac{R_n}{n}=0$. For a sufficiently large
$n$, we can construct a convex body $K\subset \mathbb{R}^n$ with $B_2^n$ is the maximal volume ellipsoid lies inside $K$ such that there
is no $P$, polytope with a polynomial number of facets in $n$ such that $K\subset P\subset R_nK$.

This result is related to approximating convex bodies by polytopes with few facets the sense of Banach-Mazur distance. In particular, it indicates that the center of John ellipsoid is not a good choice of center (of scaling) comparing to the center of mass. Speaker(s): Han Huang (UM)

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