# High-dimensional tennis balls

In this talk I will explain what a high-dimensional tennis ball is, how one can construct it and give a motivation why such an object might be of interest by connecting it to V. Milman’s question: Let $C>1$ and $\e>0$ be constants and let $k$ be an integer. Is it true that for sufficiently large $N$, every normed space $X$ that is $C$-equivalent to $\ell_2N$ has a $k$-dimensional subspace that is $(1+\e)$-complemented and $(1+\e)$-equivalent to $\ell_2k$? This is joint work with W.T. Gowers.

This talk is part of the Cambridge Analysts' Knowledge Exchange series.