Aproximación aleatoria de cuerpos convexos

Affentranger, Fernando

doi:10.5565/publmat_36192_08

Cited by 11 publications

(11 citation statements)

References 27 publications

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“…and since for all boundary points of K the principal curvatures k i are bounded from below and above by positive constants, the same holds for d 2 …”

Section: Random Points 2253mentioning

confidence: 78%

“…If K is sufficiently smooth even more can be proved: we extend (2) to an asymptotic expansion for V i (K) − E n (V i ) as n → ∞. 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 ):…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Random points on the boundary of smooth convex bodies

Reitzner

2002

Trans. Amer. Math. Soc.

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“…and since for all boundary points of K the principal curvatures k i are bounded from below and above by positive constants, the same holds for d 2 …”

Section: Random Points 2253mentioning

confidence: 78%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Random points on the boundary of smooth convex bodies

Reitzner

2002

Trans. Amer. Math. Soc.

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“…It is beyond the scope of this paper to list the results on expected values. Surveys were given by Affentranger [1], Buchta [10], Gruber [19], Schneider [39], [40], and Weil and Wieacker [43]. Many references are also contained in [5], [13], and [14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An Identity Relating Moments of Functionals of Convex Hulls

Buchta

2004

Discrete Comput Geom

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Denote by K n the convex hull of n independent random points distributed uniformly in a convex body K in R d , by V n the volume of K n , by D n the volume of K \K n , and by N n the number of vertices of K n . A well-known identity due to Efron relates the expected volume ED n -and thus EV n -to the expected number EN n+1 . This identity is extended from expected values to higher moments.The planar case of the arising identity for the variances provides in a simple way the corrected version of a central limit theorem for D n by Cabo and Groeneboom (K being a convex polygon) and an improvement of a central limit theorem for D n by Hsing (K being a circular disk). Estimates of varD n (K being a two-dimensional smooth convex body) and varN n (K being a d-dimensional smooth convex body, d ≥ 4) are obtained.The identity for moments of arbitrary order shows that the distribution of N n determines EV n−1 , EV 2 n−2 , . . . , EV n−d−1 d+1 . Reversely it is proved that these n −d −1 moments determine the distribution of N n entirely. The resulting formula for the probability that N n = k (k = d + 1, . . . , n) appears to be new for k ≥ d + 2 and yields an answer to a question raised by Baryshnikov. For k = d + 1 the formula reduces to an identity which has been repeatedly pointed out.

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“…For more information about the convex hull of random points see in particular the books by Mathai [17] and Schneider and Weil [23], as well as the surveys by Affentranger [1], Bárány [2], Buchta [5], Gruber [14], Reitzner [18], Schneider [20], [21], [22], and Weil and Wieacker [24]. Many references are also contained in [3], [7], [11], and [12].…”

Section: Introductionmentioning

confidence: 98%

On the boundary structure of the convex hull of random points

Buchta¹

2012

Advances in Geometry

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It is well known that the vertices of the convex hull of n random points, which are chosen independently and uniformly from the interior of a convex polygon, are concentrated in the neighbourhoods of the vertices of the polygon as n tends to infinity. Here concentration means that the number of vertices of the convex hull outside of the neighbourhoods of the vertices of the polygon is negligible asymptotically. The first moment of the number of vertices of the convex hull in a neighbourhood of a vertex of the polygon was obtained in 1963 by Rényi and Sulanke in a classical paper. The second moment was achieved in 1988 by Groeneboom using a Poisson point process approximation technique. Due to the complexity of the occurring calculations the extension of this technique to higher moments appears to be out of reach. Based on a purely geometric approach, which avoids stochastic processes, we derive the moments of all orders.

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Aproximación aleatoria de cuerpos convexos

Cited by 11 publications

References 27 publications

Random points on the boundary of smooth convex bodies

Random points on the boundary of smooth convex bodies

An Identity Relating Moments of Functionals of Convex Hulls

On the boundary structure of the convex hull of random points

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