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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 \(O(h^{\frac{1}{2}})\) , and the bound is tight.

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Geometry of Gauss Digitized Convex Shapes

  • 会议论文

  • 发表日期：

    2026年

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  • Jacques-Olivier Lachaud,
  • David Coeurjolly,
  • Tristan Roussillon

摘要

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 \(O(h^{\frac{1}{2}})\) , and the bound is tight.