| Popis: |
Let K be an isotropic convex body in R n and let Z q ( K ) be the L q -centroid body of K. For every N > n consider the random polytope K N : = conv { x 1 , … , x N } where x 1 , … , x N are independent random points, uniformly distributed in K. We prove that a random K N is “asymptotically equivalent” to Z [ ln ( N / n ) ] ( K ) in the following sense: there exist absolute constants ρ 1 , ρ 2 > 0 such that, for all β ∈ ( 0 , 1 2 ] and all N ⩾ N ( n , β ) , one has: (i) K N ⊇ c ( β ) Z q ( K ) for every q ⩽ ρ 1 ln ( N / n ) , with probability greater than 1 − c 1 exp ( − c 2 N 1 − β n β ) . (ii) For every q ⩾ ρ 2 ln ( N / n ) , the expected mean width E [ w ( K N ) ] of K N is bounded by c 3 w ( Z q ( K ) ) . As an application we show that the volume radius | K N | 1 / n of a random K N satisfies the bounds c 4 ln ( 2 N / n ) n ⩽ | K N | 1 / n ⩽ c 5 L K ln ( 2 N / n ) n for all N ⩽ exp ( n ) . |
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