Entropic versions of Bergström's and Bonnesen's inequalities

We establish analogues of the Bergström and Bonnesen inequalities, related to determinants and volumes respectively, for the entropy power and for the Fisher information. The obtained inequalities strengthen the well-known convolution inequality for the Fisher information as well as the entropy power inequality in dimensions $d>1$, while they reduce to the former in $d=1$. Our results recover the original Bergström inequality and generalize a proof of Bergström’s inequality given by Dembo, Cover and Thomas. We characterize the equality case in our entropic Bonnesen inequality.