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." Biometrika52, 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. Monthly76, 286-288, 1969.Peyerimhoff, N. "Areas and Intersections in Convex Domains." Amer. Math. Monthly104, 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 ." 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. Monthly101, 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 Random Points are in a Convex Position." Discrete Comput. Geom.13, 637-643, 1995.Valtr, P. "The Probability that Random Points in a Triangle are in Convex Position." Combinatorica16, 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.
，即在平面區域 
中隨機選擇的四個點，其 凸包 是一個 四邊形 (Sylvester 1865)。根據從無限平面中選取點的方法不同，可能存在許多不同的解，促使西爾維斯特得出結論：“這個問題不承認確定的解”（Sylvester 1865；Pfiefer 1989）。
是區域 
上三角形的期望面積，
是區域 
的面積 (Efron 1965)。請注意，
只是為適當區域計算的值，例如，圓盤三角形選取、三角形三角形選取、正方形三角形選取 等，其中 
可以使用 Alikoski 公式精確計算 多邊形三角形選取 的值。
可以介於
) 之間，具體取決於區域的形狀，正如 Blaschke 首次證明的那樣 (Blaschke 1923, Peyerimhoff 1997)。下表給出了各種簡單平面區域的機率 (Kendall and Moran 1963; Pfiefer 1989; Croft et al. 1991, pp. 54-55; Peyerimhoff 1997)。
中隨機選擇的 
個點的 凸包 具有 
個頂點。解由下式給出
來說是最大可能的。前幾個值是
中隨機選擇的 
個點是凸 
-邊形的頂點。解是![P_n=(2^n(3n-3)!)/([(n-1)!]^3(2n)!)](/images/equations/SylvestersFour-PointProblem/NumberedEquation4.svg)
![P_n=[1/(n!)(2n-2; n-1)]^2](/images/equations/SylvestersFour-PointProblem/NumberedEquation5.svg)
." Israel J. Math. 10, 160-171, 1971.Santaló, L. A. 
Random Points are in a Convex Position." Discrete Comput. Geom. 13, 637-643, 1995.Valtr, P. "The Probability that 
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