On the mean width of random polytopes

Random points on the boundary of smooth convex bodies

“…For i = 1 and i = d asymptotic formulae for V i (K) − E n (V i ) as in Theorem 1 are already known: for K ∈ K 3 + Buchta, Müller and Tichy [6] (for points chosen uniformly from ∂K) and Müller [13] (for arbitrary densities d K (x)) determined the asymptotic behaviour of V 1 (K) − E n (V 1 ):…”

Asymptotic estimates for best and stepwise approximation of convex bodies III

“…A sequence satisfying (1.10) is easy to construct: consider finitely many squares (i.e. cubes of dimension d-~ 1) such that the interiors of their parallel projections on bd C cover bd C. In each square choose a well-dispersed sequence of points äs deseribed by Niederreiter [18]. Atttttge the images of the points of these sequences in bd C into a sequence in the followiög way: take first the images of the first points of these sequences in a defciite order» tfaen take the images of all second points in the same order, then take the images of aU third points, again in the same order, etc.…”

Aproximación aleatoria de cuerpos convexos

“…Por otro lado, si los n puntos están distribuidos de manera aleatoria sobre 8K ,y su densidad de distribución es g(s) = K(s) 2 ic(s) ds ax (es decir, la ley de distribución depende de la curvatura de OK) tenemos que (4) L(K) -E(L(H..))^' 4 ( f r, (S) ds)s n , n -> oo, ax siendo E(L(Hn )) el valor medio de L(H) (cf. [44] ).…”

“…[20], [1] y [45]), -n puntos aleatorios sobre un cuerpo convexo suave que admiten una distribución absolutarnente continua respecto de la medida de superficie de &K y cuya densidad de distribución cumple ciertos requisitos de continuidad (cf. [44]), -n puntos aleatorios de los cuales n -i están distribuidos de manera uniforme; en la d-esfera Bd y los restantes i de manera uniforme; sobre el borde de Bd (cf. [43], [1]) .…”