Properties And Conditions Of Sets With Positive Reach

The reach of a subset in Euclidean space is an important geometric no-tion introduced by H.Federer,which has important applications in geometry and analysis. In this paper,we first compute the reach of some special curves.and then give a specific formula for the reach of general C2 smooth curves. Esti-mating the volume(Hausdorff measure)of a s-neighborhood of a closed subset in Euclidean space is an interesting geometric problem. Another focus of this paper is to study the inequality relation between the n-dimensional Hausdorff measure of s-neighborhood of A and the (n-1)-dimensional Hausdorff measure for a subset. A with positive reach in Rn. In order to establish the inequality relation,we use two key facts:estimating the Lipschitz constant of the unique nearest point map and Eilenberg inequality.