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Relative to the Gaussian measure on $\mathbb{R}^d$, entropy and Fisherinformation 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 interpolatesamong these relations. Several interesting corollaries follow: (i) the deficitin the LSI satisfies a convolution inequality itself; (ii) the deficit in theLSI controls convergence in the entropic and Fisher information central limittheorems; and (iii) the LSI is stable with respect to HWI jumps (i.e., a jumpin any of the convolution inequalities associated to the HWI functionals).  Another consequence is that the convolution inequalities for Fisherinformation and entropy powers are reversible in general, up to a factordepending on the Stam defect. An improved form of Nelson's hypercontractivityestimate also follows. Finally, we speculate on the possibility of an analogousreverse Brunn-Minkowski inequality and a related upper bound on surface areaassociated to Minkowski sums.

Links between the Logarithmic Sobolev Inequality and the convolution inequalities for Entropy and Fisher Information