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 on X, 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 \(O(h^{\frac{2d}{d+1}})\) ). Our main result states 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. Finally we compare experimentally the performances of several normal estimators built upon the normal vectors to the facets of \(Y_h\) with state-of-the-art estimators. We also perform statistical analyses over the facets of digitized convex hulls, like their area, diameter or width as a function of the digitization gridstep. Both our new theoretical properties and our numerical experiments confirm that the convex hull of a digitized shape provide relevant information on the geometry of the underlying Euclidean convex shape, and can be used to construct fast and accurate geometric estimators.
