  Let $M\subset {\mathbf R}^{m+1}$ be a smooth, closed, codimension-one
self-shrinker (for mean curvature flow) with nontrivial $k^{\rm th}$ homology.
We show that the entropy of $M$ is greater than or equal to the entropy of a
round $k$-sphere, and that if equality holds, then $M$ is a round $k$-sphere in
${\mathbf R}^{k+1}$.
