A Sharp Conservative Offset for Marching Cubes

Given a grid-sampled 1-Lipschitz function $f:{\mathbb{R}}^3\to \mathbb{R}$ (such as a signed distance function) sampled on a regular grid, we determine the sharp infimum ${\sigma }_{\star }$ such that for every $\sigma >{\sigma }_{\star }$ the true zero-level set $f^{-1}(0)$ is contained in the marching cubes mesh for $f=\sigma$. For grids with spacing $h$, the sharp infimum is ${\sigma }_{\star }=\frac{\sqrt{3}}{2}h$. This result is proven \emph{despite} the planar marching cube faces potentially lying on either side of the level set of the trilinearly interpolated approximation of $f$ (which shares the same infimum). The proof instead constructs a convex lift of the corner samples and shows that the high-side portion of each marching-cubes cell lies in the projection of its $\sigma$-superlevel portion, which cannot contain a point of $f^{-1}(0)$.