Geometry of Gauss digitized convex shapes

This paper studies how well we can infer the geometry of a (smooth or not) convex shape $X$ from the convex hull $Y_h$ of its Gauss digitization with a given gridstep $h$. Without smoothness constraint, we first present results concerning the proximity of facet normal vectors to the shape normal vectors, as well as a relation between the number of lattice points just above a facet and its area. Then, further results can be obtained when $X$ is smooth, that are valid in arbitrary dimension $d$. More precisely, we show that the boundary of $Y_h$ is Hausdorff-close to the boundary of $X$ with distance less than $\sqrt{d}h$, and that the vertices of $Y_h$ are even much closer (some $O(h^{\frac{2d}{d+1}})$). Finally we show that the geometric normal vectors to the facets of $Y_h$ tend to the smooth shape normals with a speed $O(\sqrt{h})$, and the bound is tight.

Discrete Geometry and Mathematical Morphology - 4th International Joint Conference, {DGMM} 2025, Groningen, The Netherlands, November 3-6, 2025, Proceedings, volume 16296 of Lecture Notes in Computer Science, pp 16–30, 2025. Springer