Compact families of Jordan curves and convex hulls in three dimensions
CÓ Dúnlaing - arXiv preprint arXiv:1311.6331, 2013 - arxiv.org
CÓ Dúnlaing
arXiv preprint arXiv:1311.6331, 2013•arxiv.orgWe prove that for certain families of semi-algebraic convex bodies in 3 dimensions, the
convex hull of $ n $ disjoint bodies has $ O (n\lambda_s (n)) $ features, where $ s $ is a
constant depending on the family: $\lambda_s (n) $ is the maximum length of order-$ s $
Davenport-Schinzel sequences with $ n $ letters. The argument is based on an apparently
new idea ofcompact families' of convex bodies or discs, and ofcrossing content'of disc
intersections.
convex hull of $ n $ disjoint bodies has $ O (n\lambda_s (n)) $ features, where $ s $ is a
constant depending on the family: $\lambda_s (n) $ is the maximum length of order-$ s $
Davenport-Schinzel sequences with $ n $ letters. The argument is based on an apparently
new idea ofcompact families' of convex bodies or discs, and ofcrossing content'of disc
intersections.
We prove that for certain families of semi-algebraic convex bodies in 3 dimensions, the convex hull of disjoint bodies has features, where is a constant depending on the family: is the maximum length of order- Davenport-Schinzel sequences with letters. The argument is based on an apparently new idea of `compact families' of convex bodies or discs, and of `crossing content' of disc intersections.
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