论文标题
D维多型中的随机体积
Random volumes in d-dimensional polytopes
论文作者
论文摘要
假设我们从$ d $尺寸中的凸件中随机选择$ n $点。 $ n $在$ d $方面必须有多大，以便点的凸壳几乎与凸尸体本身一样大？ Dyer-füredi-McDiarmid显示了当凸体是HyperCube时，许多样本就足够了，而Pivovarov则表明欧几里得球要求大约$ d^{d/2} $样本。我们表明，当凸体是单纯形时，许多样本就足够了。然后，这意味着对于任何凸的简单多型的结果都具有相同的结果。
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.
