Geodesic
Random volumes in d-dimensional polytopes
Discrete analysis (2020)
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Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Füredi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.
Publié le : 
@article{DAS_2020_a5,
     author = {Alan Frieze and Wesley Pegden and Tomasz Tkocz},
     title = {Random volumes in d-dimensional polytopes},
     journal = {Discrete analysis},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2020_a5/}
}
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AU  - Tomasz Tkocz
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JO  - Discrete analysis
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%A Alan Frieze
%A Wesley Pegden
%A Tomasz Tkocz
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Alan Frieze; Wesley Pegden; Tomasz Tkocz. Random volumes in d-dimensional polytopes. Discrete analysis (2020). http://geodesic.mathdoc.fr/item/DAS_2020_a5/