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西尔维斯特四点问题

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SylvestersFourPoints

西尔维斯特四点问题询问的是概率 q(R)，即在平面区域 R 中随机选择的四个点，其 凸包 是一个 四边形 (Sylvester 1865)。根据从无限平面中选取点的方法不同，可能存在许多不同的解，促使西尔维斯特得出结论：“这个问题不承认确定的解”（Sylvester 1865；Pfiefer 1989）。

对于从平面 平面 的具有有限 面积 的开凸子集中选取的点，概率由下式给出

P(R)=1-(4A^__R)/(A(R)),
(1)

其中 A^__R 是区域 R 上三角形的期望面积，A(R) 是区域 R 的面积 (Efron 1965)。请注意，A^__R 只是为适当区域计算的值，例如，圆盘三角形选取、三角形三角形选取、正方形三角形选取 等，其中 A_R 可以使用 Alikoski 公式精确计算 多边形三角形选取 的值。

P(R) 可以介于

2/3<=q(R)<=1-(35)/(12pi^2)
(2)

(0.66666<=q(R)<=0.70448) 之间，具体取决于区域的形状，正如 Blaschke 首次证明的那样 (Blaschke 1923, Peyerimhoff 1997)。下表给出了各种简单平面区域的概率 (Kendall and Moran 1963; Pfiefer 1989; Croft et al. 1991, pp. 54-55; Peyerimhoff 1997)。

RP(R)约
三角形2/30.66667
正方形(25)/(36)0.69444
五边形2/(45)(18-sqrt(5))0.70062
六边形(683)/(972)0.70267
椭圆, 圆盘1-(35)/(12pi^2)0.70448

西尔维斯特问题可以推广为询问概率，即在 单位球 B^n 中随机选择的 n+2 个点的 凸包 具有 n+1 个顶点。解由下式给出

P_n=((n+2)(n+1; 1/2(n+1))^(n+1))/(2^n((n+1)^2; 1/2(n+1)^2))
(3)

(Kingman 1969, Groemer 1973, Peyerimhoff 1997)，这对于任何有界的 凸域 K in R^n 来说是最大可能的。前几个值是

P_1=1
(4)
P_2=(35)/(12pi^2)
(5)
P_3=9/(143)
(6)
P_4=(676039)/(648000pi^4)
(7)
P_5=(20000)/(12964479)
(8)

(OEIS A051050 和 A051051)。

另一个推广询问的是概率，即在固定的有界 凸域 K subset R^2 中随机选择的 n 个点是凸 n-边形的顶点。解是

P_n=(2^n(3n-3)!)/([(n-1)!]^3(2n)!)
(9)

对于三角形域，其前几个值为 1, 1, 1, 2/3, 11/36, 91/900, 17/675, ... (OEIS A004677 和 A004824)，以及

P_n=[1/(n!)(2n-2; n-1)]^2
(10)

对于平行四边形域，其前几个值为 1, 1, 1, 25/36, 49/144, 121/3600, ... (OEIS A004936 和 A005017; Valtr 1996, Peyerimhoff 1997)。

西尔维斯特四点问题与图的 直线交叉数 有着意想不到的联系 (Finch 2003)。

另请参阅

圆盘三角形选取, 六边形三角形选取, 多边形三角形选取, 直线交叉数, 正方形三角形选取, 三角形三角形选取

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参考文献

Alikoski, H. A. "Über das Sylvestersche Vierpunktproblem." Ann. Acad. Sci. Fenn. 51, No. 7, 1-10, 1939.Blaschke, W. "Über affine Geometrie XI: Lösung des 'Vierpunktproblems' von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten." Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Phys. Kl. 69, 436-453, 1917.Blaschke, W. §24-25 in Vorlesungen über Differentialgeometrie, II. Affine Differentialgeometrie. Berlin: Springer-Verlag, 1923.Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Random Polygons and Polyhedra." §B5 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 54-57, 1991.Crofton, M. W. "Probability." Encyclopedia Britannica, Vol. 19, 9th ed. pp. 768-788, 1885.Efron, B. "The Convex Hull of a Random Set of Points." Biometrika 52, 331-343, 1965.Finch, S. R. "Rectilinear Crossing Constant." §8.18 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 532-534, 2003.Groemer, H. "On Some Mean Values Associated with a Randomly Selected Simlpex in a Convex Set." Pacific J. Math. 45, 525-533, 1973.Kendall, M. G. and Moran, P. A. P. Geometrical Probability. New York: Hafner, 1963.Kingman, J. F. C. "Random Secants of a Convex Body." J. Appl. Prob. 6, 660-672, 1969.Klee, V. "What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body." Amer. Math. Monthly 76, 286-288, 1969.Peyerimhoff, N. "Areas and Intersections in Convex Domains." Amer. Math. Monthly 104, 697-704, 1997.Pfiefer, R. E. "The Historical Development of J. J. Sylvester's Four Point Problem." Math. Mag. 62, 309-317, 1989.Rottenberg, R. R. "On Finite Sets of Points in P^3." Israel J. Math. 10, 160-171, 1971.Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.Scheinerman, E. and Wilf, H. S. "The Rectilinear Crossing Number of a Complete Graph and Sylvester's 'Four Point' Problem of Geometric Probability." Amer. Math. Monthly 101, 939-943, 1994.Sloane, N. J. A. Sequences A004677, A004824, A004936, A005017, A051050, and A051051 in "The On-Line Encyclopedia of Integer Sequences."Solomon, H. "Crofton's Theorem and Sylvester's Problem in Two and Three Dimensions." Ch. 5 in Geometric Probability. Philadelphia, PA: SIAM, pp. 97-125, 1978.Sylvester, J. J. "Question 1491." The Educational Times (London). April 1864.Sylvester, J. J. "On a Special Class of Questions on the Theory of Probabilities." Birmingham British Assoc. Rept., pp. 8-9, 1865.Valtr, P. "Probability that n Random Points are in a Convex Position." Discrete Comput. Geom. 13, 637-643, 1995.Valtr, P. "The Probability that n Random Points in a Triangle are in Convex Position." Combinatorica 16, 567-573, 1996.Weil, W. and Wieacker, J. "Stochastic Geometry." Ch. 5.2 in Handbook of Convex Geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 1391-1438, 1993.Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In Combinatorics, Geometry, and Probability: A Tribute to Paul Erdős. Papers from the Conference in Honor of Erdős' 80th Birthday Held at Trinity College, Cambridge, March 1993 (Ed. B. Bollobás and A. Thomason). Cambridge, England: Cambridge University Press, pp. 557-562, 1997.Woolhouse, W. S. B. "Some Additional Observations on the Four-Point Problem." Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 7. London: F. Hodgson and Son, p. 81, 1867.

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西尔维斯特四点问题

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