The John-Nirenberg inequality with sharp constants

We consider the one-dimensional John-Nirenberg inequality: $$ |\{x\in I_0:|f(x)-f_{I_0}|>\a\}|\le C_1|I_0|\exp\Big(-\frac{C_2}{\|f\|_{*}}\a\Big). $$ A. Korenovskii found that the sharp $C_2$ here is $C_2=2/e$. It is shown in this paper that if $C_2=2/e$, then the best possible $C_1$ is $C_1= \frac{1}{2}e^{4/e}$.