I just study a paper by Saloff-Coste
http://matwbn.icm.edu.pl/ksiazki/cm/cm67/cm67111.pdf
On page 118 there is theorem 9
And on page 119 in remark 3, there is an inequality
The whole thing that I want to know is about $C(n, k, \epsilon)$.
$C$ is a constant that depends to dimension $n$. But what order of $n$? If $C=O(n^x)$ where $x>0$ (i.e. $C$ tends to infinity as $n$ grows up) my problem will be solved.
I will be very grateful if I have your point of view in this case. Thanks