New Lower Bounds for Convex Hull Problems in Odd Dimensions

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New Lower Bounds for Convex Hull Problems in Odd Dimensions (1996)

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by Jeff Erickson

Venue:	SIAM J. Comput
Citations:	32 - 6 self

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@ARTICLE{Erickson96newlower,
    author = {Jeff Erickson},
    title = {New Lower Bounds for Convex Hull Problems in Odd Dimensions},
    journal = {SIAM J. Comput},
    year = {1996},
    volume = {28},
    pages = {1--9}
}

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Abstract

We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is &lceil;d/2&rceil;-hard, in the in the sense of Gajentaan and Overmars.

Keyphrases

odd dimension new lower bound convex hull problem several year quasi-simplicial n-vertex polytope key step sidedness query similar technique degenerate facet convex hull facet upper bound related result d-dimensional convex hull correct proof straightforward adversary argument arbitrary dimension simplicial convex hull circular degeneracy affine degeneracy convex hull bound match

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