Poincare's inequalities and Talagrand's concentration phenomenon for the exponential distribution

We present a simple proof, based oil modified logarithmic Sobolev inequalities, of Talagrand's concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincare inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps.