Conic reach and polynomial parallel volume in the plane

What the math gives to ML

The paper gives an explicit local formula for the area of a small tubular neighborhood around planar sets that are exactly conic near finitely many singularities and have a uniform lower bound on normal-fiber lengths elsewhere. The transferable asset is not the reach notion itself, but the fact that local geometric complexity is detectable through the coefficients and residuals of a low-degree tube-volume curve. A practical neural-network use is to regularize a two-dimensional latent embedding or decision-boundary projection so that its small-scale neighborhood growth is approximately quadratic, suppressing cusps, tangential contacts, and increasingly narrow gaps that can harm robustness and interpolation. This is most credible as an auxiliary loss on low-dimensional projections rather than as a general high-dimensional theorem.