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  • English  (8)
  • 1
    Online Resource
    Online Resource
    Random projections of regular simplices
    Springer Science and Business Media LLC ; 1992
    In:  Discrete & Computational Geometry Vol. 7, No. 3 ( 1992-3), p. 219-226
    Details
    In: Discrete & Computational Geometry, Springer Science and Business Media LLC, Vol. 7, No. 3 ( 1992-3), p. 219-226
    Type of Medium: Online Resource
    ISSN: 0179-5376 , 1432-0444
    URL: Article
    DOI: 10.1007/BF02187839
    Language: English
    Publisher: Springer Science and Business Media LLC
    Publication Date: 1992
    detail.hit.zdb_id: 1459007-4
    detail.hit.zdb_id: 53957-0
    SSG: 17,1
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  • 2
    Online Resource
    Online Resource
    The convex hull of random balls
    Cambridge University Press (CUP) ; 1993
    In:  Advances in Applied Probability Vol. 25, No. 02 ( 1993-06), p. 373-394
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 25, No. 02 ( 1993-06), p. 373-394
    Abstract: While the convex hull of n d -dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω( n [ d /2] ). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d -ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d -balls is investigated for each. As n grows without bound, these expectations are O (1), O ( n ( d –1)/( d +4) ), O (1) (for d = 2 only), and O ( n ( d– 1)/( d+ 3) ).
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.1017/S0001867800025404
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1993
    detail.hit.zdb_id: 1474602-5
    detail.hit.zdb_id: 160005-9
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  • 3
    Online Resource
    Online Resource
    The convex hull of random balls
    Cambridge University Press (CUP) ; 1993
    In:  Advances in Applied Probability Vol. 25, No. 2 ( 1993-06), p. 373-394
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 25, No. 2 ( 1993-06), p. 373-394
    Abstract: While the convex hull of n d -dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω( n [ d /2] ). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d -ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d -balls is investigated for each. As n grows without bound, these expectations are O (1), O ( n ( d –1)/( d +4) ), O (1) (for d = 2 only), and O ( n ( d– 1)/( d+ 3) ).
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.2307/1427658
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1993
    detail.hit.zdb_id: 1474602-5
    detail.hit.zdb_id: 160005-9
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  • 4
    Online Resource
    Online Resource
    The expected volume of a random polytope in a ball*
    Wiley ; 1988
    In:  Journal of Microscopy Vol. 151, No. 3 ( 1988-09), p. 277-287
    Details
    In: Journal of Microscopy, Wiley, Vol. 151, No. 3 ( 1988-09), p. 277-287
    Abstract: For any convex body K in d ‐dimensional Euclidean space E d ( d ≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n , let V ( d ) n‐i i(K ) be the expected volume of the convex hull H n‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K . We develop an integral formula for V ( d ) n‐i, i ( K ) for the case that K is a d ‐dimensional unit ball by considering an adequate decomposition of V ( d ) n‐i , i into d ‐dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of B d , we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three‐dimensional cases.
    Type of Medium: Online Resource
    ISSN: 0022-2720 , 1365-2818
    URL: Issue
    URL: Article
    DOI: 10.1111/jmi.1988.151.issue-3
    DOI: 10.1111/j.1365-2818.1988.tb04688.x
    Language: English
    Publisher: Wiley
    Publication Date: 1988
    detail.hit.zdb_id: 219263-9
    detail.hit.zdb_id: 2007259-4
    SSG: 11
    SSG: 12
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  • 5
    Online Resource
    Online Resource
    On the convex hull of uniform random points in a simpled-polytope
    Springer Science and Business Media LLC ; 1991
    In:  Discrete & Computational Geometry Vol. 6, No. 3 ( 1991-9), p. 291-305
    Details
    In: Discrete & Computational Geometry, Springer Science and Business Media LLC, Vol. 6, No. 3 ( 1991-9), p. 291-305
    Type of Medium: Online Resource
    ISSN: 0179-5376 , 1432-0444
    URL: Article
    DOI: 10.1007/BF02574691
    Language: English
    Publisher: Springer Science and Business Media LLC
    Publication Date: 1991
    detail.hit.zdb_id: 1459007-4
    detail.hit.zdb_id: 53957-0
    SSG: 17,1
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  • 6
    Online Resource
    Online Resource
    Random spheres in a convex body
    Springer Science and Business Media LLC ; 1990
    In:  Archiv der Mathematik Vol. 55, No. 1 ( 1990-7), p. 74-81
    Details
    In: Archiv der Mathematik, Springer Science and Business Media LLC, Vol. 55, No. 1 ( 1990-7), p. 74-81
    Type of Medium: Online Resource
    ISSN: 0003-889X , 1420-8938
    URL: Article
    DOI: 10.1007/BF01199118
    RVK:
    SA 2760
    Language: English
    Publisher: Springer Science and Business Media LLC
    Publication Date: 1990
    detail.hit.zdb_id: 1458441-4
    detail.hit.zdb_id: 475-3
    SSG: 17,1
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  • 7
    Online Resource
    Online Resource
    Random circles in the d-dimensional unit ball
    Cambridge University Press (CUP) ; 1989
    In:  Journal of Applied Probability Vol. 26, No. 02 ( 1989-06), p. 408-412
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 26, No. 02 ( 1989-06), p. 408-412
    Abstract: This note gives the solution of the following problem concerning geometric probabilities. What is the probability p ( B d ; 2) that the circumference determined by three points P, P 1 and P 2 chosen independently and uniformly at random in the interior of a d -dimensional unit ball B d in Euclidean space E d ( d ≧ 2) is entirely contained in B d ? From our result we conclude that p ( B d ; 2) →π /(3√3) as d →∞.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200027406
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1989
    detail.hit.zdb_id: 219147-7
    detail.hit.zdb_id: 1474599-9
    SSG: 3,2
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  • 8
    Online Resource
    Online Resource
    Random circles in the d -dimensional unit ball
    Cambridge University Press (CUP) ; 1989
    In:  Journal of Applied Probability Vol. 26, No. 2 ( 1989-06), p. 408-412
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 26, No. 2 ( 1989-06), p. 408-412
    Abstract: This note gives the solution of the following problem concerning geometric probabilities. What is the probability p ( B d ; 2) that the circumference determined by three points P, P 1 and P 2 chosen independently and uniformly at random in the interior of a d -dimensional unit ball B d in Euclidean space E d ( d ≧ 2) is entirely contained in B d ? From our result we conclude that p ( B d ; 2) →π /(3√3) as d →∞.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.2307/3214047
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1989
    detail.hit.zdb_id: 219147-7
    detail.hit.zdb_id: 1474599-9
    SSG: 3,2
    Location Call Number Limitation Availability
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