A polynomial bound in Dvoretzky's theorem

We present a simple proof of the $\epsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\epsilon$ in Dvoretzky’s theorem is polynomial in $1\mathrm{/}\epsilon$. In particular, if $n\ge (C\mathrm{/}\epsilon {)}^{\mathrm{\ell }\mathrm{/}2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\mathrm{\ell }$-dimensional section that is $\epsilon$-close to a Euclidean ball. Here, $C>0$ is a universal constant. We in fact obtain a sharper dependence on $\epsilon$. 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.