Capacitary-Distance Hardy Inequality

What the math gives to ML

The paper provides a sharp Hardy inequality in which ordinary distance to a boundary is replaced by a capacitary distance that detects thin or highly fragmented obstacles. Its transferable asset is a computable coercivity principle: penalizing a function by the inverse squared capacitary distance is controlled by its Dirichlet energy, with explicit and sharp alpha^{-2} dependence. This suggests a boundary-aware regularizer for neural fields or coordinate MLPs defined on domains containing holes, masks, or disconnected forbidden regions, where Euclidean distance can badly underestimate geometric difficulty. The practical route is to estimate capacitary distance on a voxel or point-cloud proxy and add the theorem-inspired weighted value penalty during training.