Quantitative Stability of Brunn-Minkowski Inequalities for the p-Torsional Rigidity

We establish the quantitative Brunn-Minkowski inequality and the quantitative Urysohn inequality for p-torsional rigidity. Here the p-torsional rigidity can be formulated by the weak solution of the boundary value problem of p-Laplace equation. In order to prove the main results, we establish the relation between the Borell-Brascamp-Lieb deficit and the Brunn-Minkowski deficit, and then use the relation to prove two different stability of the Borell-Brascamp-Lieb inequality for p-concave functions with p > 0. The constants in our stability results are independent on the parameter p and the integrals of given functions, and thus also improve the results given by Ghilli and Salani about the case p = 2.