Sharp dependence on the interpolation parameter

Determine whether the stability estimate O_{n,\lambda,p}(\sqrt{\delta}) for the Borell–Brascamp–Lieb inequality can be replaced by the sharper bound O_{n,p}(\sqrt{\delta/\lambda}), thereby establishing the conjectured dependence on the parameter \lambda.

Background

The paper proves quantitative stability for the Borell–Brascamp–Lieb inequality with an error term of order O_{n,\lambda,p}(\sqrt{\delta}), where \delta measures the deficit and \lambda is the interpolation parameter. The authors note that corresponding sharp stability results for the Brunn–Minkowski inequality suggest the more precise dependence O_{n,p}(\sqrt{\delta/\lambda}).

Establishing this improvement would require additional nontrivial arguments beyond those developed in the paper, so the sharper dependence remains unresolved and is explicitly left for future work. The problem is included because it concerns the optimal quantitative dependence of the stability estimate on a fundamental parameter of the inequality.