  We establish a lower bound on the entropy of weighted sums of (possibly
dependent) random variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric
joint distribution. Our lower bound is in terms of the joint entropy of $(X_1,
X_2, \dots, X_n)$. We show that for $n \geq 3$, the lower bound is tight if and
only if $X_i$'s are i.i.d.\ Gaussian random variables. For $n=2$ there are
numerous other cases of equality apart from i.i.d.\ Gaussians, which we
completely characterize. Going beyond sums, we also present an inequality for
certain linear transformations of $(X_1, \dots, X_n)$. Our primary technical
contribution lies in the analysis of the equality cases, and our approach
relies on the geometry and the symmetry of the problem.
