Non-squeezing property of contact balls

In this paper we solve a contact non-squeezing conjecture proposed by Eliashberg, Kim and Polterovich. Let $B_R$ be the open ball of radius $R$ in ${\mathbf{R}}^{2n}$ and let ${\mathbf{R}}^{2n}\times {\mathbf{S}}^1$ be the prequantization space equipped with the standard contact structure. Following Tamarkin’s idea, we apply microlocal category methods to prove that if $R$ and $r$ satisfy $1\le \pi r^2<\pi R^2$, then it is impossible to squeeze the contact ball $B_R\times {\mathbf{S}}^1$ into $B_r\times {\mathbf{S}}^1$ via compactly supported contact isotopies.