The John-Nirenberg inequality with sharp constants

We consider the one-dimensional John-Nirenberg inequality: $$\mathrm{\mid }\{x\in I_0:\mathrm{\mid }f(x)-f_{I_0}\mathrm{\mid }>\text{a}\}\mathrm{\mid }\le C_1\mathrm{\mid }{I}_0\mathrm{\mid }\exp (-\frac{C_2}{\mathrm{\parallel }f{\mathrm{\parallel }}_{\ast }}\text{a})\mathrm{.}$$ A. Korenovskii found that the sharp $C_2$ here is $C_2=2\mathrm{/}e$. It is shown in this paper that if $C_2=2\mathrm{/}e$, then the best possible $C_1$ is $C_1= \frac{1}{2}e^{4/e}$.