A polynomial bound in Dvoretzky's theorem
We present a simple proof of the $\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\varepsilon$. In particular, if $n \geq (C/\varepsilon)^{\ell/2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\ell$-dimensional section that is $\varepsilon$-close to a Euclidean ball. Here, $C > 0$ is a universal constant. We in fact obtain a sharper dependence on $\varepsilon$. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.