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The entropy power inequality states that the effective variance (entropy power) of the sum of two independent random variables is greater than the sum of their effective variances. The Brunn-Minkowski inequality states that the effective radius of the set sum of two sets is greater than the sum of their effective radii. Both these inequalities are recast in a form that enhances their similarity. In spite of this similarity, there is as yet no common proof of the inequalities. Nevertheless, their intriguing similarity suggests that new results relating to entropies from known results in geometry and vice versa may be found. Two applications of this reasoning are presented. First, an isoperimetric inequality for entropy is proved that shows that the spherical normal distribution minimizes the trace of the Fisher information matrix given an entropy constraint--just as a sphere minimizes the surface area given a volume constraint. Second, a theorem involving the effective radii of growing convex sets is proved.