  Relative to the Gaussian measure on $\mathbb{R}^d$, entropy and Fisher
information are famously related via Gross' logarithmic Sobolev inequality
(LSI). These same functionals also separately satisfy convolution inequalities,
as proved by Stam. We establish a dimension-free inequality that interpolates
among these relations. Several interesting corollaries follow: (i) the deficit
in the LSI satisfies a convolution inequality itself; (ii) the deficit in the
LSI controls convergence in the entropic and Fisher information central limit
theorems; and (iii) the LSI is stable with respect to HWI jumps (i.e., a jump
in any of the convolution inequalities associated to the HWI functionals).
  Another consequence is that the convolution inequalities for Fisher
information and entropy powers are reversible in general, up to a factor
depending on the Stam defect. An improved form of Nelson's hypercontractivity
estimate also follows. Finally, we speculate on the possibility of an analogous
reverse Brunn-Minkowski inequality and a related upper bound on surface area
associated to Minkowski sums.
