Summary: A lot of forms and applications have been found in Poincaré inequalities, but the optimal constants satisfying a Poincaré inequality have not been estimated. This paper estimates the optimum constants \(\lambda_0\) and \(\lambda_1\) satisfying a Poincaré inequality by using isoperimetric constants. It is \(\lambda_0\geqslant k^2_0/ (2R)\) and \(\lambda_1\geqslant {k'}^2_1/ (2R)\).