Entropic analogues of Grünbaum's inequality

What the math gives to ML

The paper supplies a sharp entropy-preservation guarantee for cutting a log-concave distribution at its mean: conditioning on either side of a mean hyperplane cannot reduce differential entropy by more than an explicit universal constant. This is potentially useful for hierarchical routers and latent partitioning, where repeated gating can otherwise create low-entropy, collapsed subpopulations. The most promising transfer is not to use the theorem as a generic entropy regularizer, but to build mean-centered binary routing with an entropy-deficit monitor or barrier, while separately enforcing a total-entropy floor. The guarantee assumes log-concavity, so experiments should begin with approximately Gaussian or Laplace latent features.