
This article is cited in 1 scientific paper (total in 1 paper)
An estimate of the Hausdorff distance between a set and its convex hull in Euclidean spaces of small dimension
V. N. Ushakov^{a}, A. A. Ershov^{ba} ^{a} Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
^{b} Chelyabinsk State University
Abstract:
We derive estimates for the Hausdorff distance between sets and their convex hulls in finitedimensional Euclidean spaces with the standard scalar product and the corresponding norm. In the first part of the paper, we consider estimates for $\alpha$sets. By an $\alpha$set we mean an arbitrary compact set for which the parameter characterizing the degree of nonconvexity and computed in a certain way equals $\alpha$. In most cases, the parameter $\alpha$ is the maximum possible angle under which the projections to this set are visible from points not belonging to the set. Note that $\alpha$sets were introduced by V.N. Ushakov for the classification of nonconvex sets according to the degree of their nonconvexity; $\alpha$sets are used for the description of wavefronts and for the solution of other problems in control theory. We consider $\alpha$sets only in a twodimensional space. It is proved that, if $\alpha$ is small, then the corresponding $\alpha$sets are close to convex sets in the Hausdorff metric. This allows to neglect their nonconvexity and consider such sets convex if it is known that the parameter $\alpha$ is small. The known ShapleyFolkman theorem is often applied in the same way. In the second part of the paper we present some improvements of the estimates from the ShapleyFolkman theorem. The original ShapleyFolkman theorem states that the Minkowski sum of a large number of sets is close in the Hausdorff metric to the convex hull of this sum with respect to the value of the Chebyshev radius of the sum. We consider a particular case when the sum consists of identical terms; i.e., we add some set $M$ to itself. For this case we derive an improved estimate, which is essential for sets in spaces of small dimension. In addition, as in Starr's known corollary, the new estimate admits the following improvement: the Chebyshev radius $R(M)$ on the righthand side can be replaced by the inner radius $r(M)$ of the set $M$. However, as the dimension of the space grows, the new estimate tends asymptotically to the estimate following immediately from the ShapleyFolkman theorem.
Keywords:
$\alpha$set, Minkowski sum, convex hull, Hausdorff distance.
DOI:
https://doi.org/10.21538/013448892018241223235
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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2019, 305, suppl. 1, S178–S190
Bibliographic databases:
UDC:
517.977
MSC: 52A27, 52A30 Received: 10.09.2017
Citation:
V. N. Ushakov, A. A. Ershov, “An estimate of the Hausdorff distance between a set and its convex hull in Euclidean spaces of small dimension”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 1, 2018, 223–235; Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S178–S190
Citation in format AMSBIB
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\pages 223235
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\jour Proc. Steklov Inst. Math. (Suppl.)
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