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<!doctype html> <html lang="en" class="geometry numbertheory"> <head> <title>Sylvester's Four-Point Problem -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Sylvester's Four-Point Problem" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="Sylvester's four-point problem asks for the probability q(R) that four points chosen at random in a planar region R have a convex hull which is a quadrilateral (Sylvester 1865). Depending on the method chosen to pick points from the infinite plane, a number of different solutions are possible, prompting Sylvester to conclude "This problem does not admit of a determinate solution" (Sylvester 1865; Pfiefer 1989). For points selected from an open, convex subset of the plane having..." /> <meta name="description" content="Sylvester's four-point problem asks for the probability q(R) that four points chosen at random in a planar region R have a convex hull which is a quadrilateral (Sylvester 1865). Depending on the method chosen to pick points from the infinite plane, a number of different solutions are possible, prompting Sylvester to conclude "This problem does not admit of a determinate solution" (Sylvester 1865; Pfiefer 1989). For points selected from an open, convex subset of the plane having..." /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2004-04-15" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Computational Geometry:Random Point Picking" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Computational Geometry:Convex Hulls" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Plane Geometry:Quadrilaterals" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Number Theory:Constants:Geometric Constants" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Geometry:Points" /> <meta name="DC.Subject" scheme="MSC_2000" content="41" /> <meta name="DC.Subject" scheme="MSC_2000" content="51M04" /> <meta name="DC.Subject" scheme="MSC_2000" content="52A" /> <meta name="DC.Subject" scheme="MSC_2000" content="65D" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. 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Depending on the method chosen to pick points from the infinite plane, a number of different solutions are possible, prompting Sylvester to conclude "This problem does not admit of a determinate solution" (Sylvester 1865; Pfiefer 1989). For points selected from an open, convex subset of the plane having..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Sylvester's Four-Point Problem -- from Wolfram MathWorld"> <meta name="twitter:description" content="Sylvester's four-point problem asks for the probability q(R) that four points chosen at random in a planar region R have a convex hull which is a quadrilateral (Sylvester 1865). Depending on the method chosen to pick points from the infinite plane, a number of different solutions are possible, prompting Sylvester to conclude "This problem does not admit of a determinate solution" (Sylvester 1865; Pfiefer 1989). 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History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/ComputationalGeometry.html">Computational Geometry</a> </li> <li> <a href="/topics/RandomPointPicking.html">Random Point Picking</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/ComputationalGeometry.html">Computational Geometry</a> </li> <li> <a href="/topics/ConvexHulls.html">Convex Hulls</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/PlaneGeometry.html">Plane Geometry</a> </li> <li> <a href="/topics/Quadrilaterals.html">Quadrilaterals</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/NumberTheory.html">Number Theory</a> </li> <li> <a href="/topics/Constants.html">Constants</a> </li> <li> <a href="/topics/GeometricConstants.html">Geometric Constants</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/Geometry.html">Geometry</a> </li> <li> <a href="/topics/Points.html">Points</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>   <h1>Sylvester's Four-Point Problem</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/ComputationalGeometry/SylvestersFour-PointProblem.nb" download="SylvestersFour-PointProblem.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="467.654" height="222.666" src="images/eps-svg/SylvestersFourPoints_700.svg" class="" alt="SylvestersFourPoints" /> </div> <p> Sylvester's four-point problem asks for the probability <img src="/images/equations/SylvestersFour-PointProblem/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="34" height="21" alt="q(R)" /> that four points chosen at random in a planar region <img src="/images/equations/SylvestersFour-PointProblem/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="R" /> have a <a href="/ConvexHull.html">convex hull</a> which is a <a href="/Quadrilateral.html">quadrilateral</a> (Sylvester 1865). Depending on the method chosen to pick points from the infinite plane, a number of different solutions are possible, prompting Sylvester to conclude "This problem does not admit of a determinate solution" (Sylvester 1865; Pfiefer 1989). </p> <p> For points selected from an open, convex subset of the <a href="/Plane.html">plane</a> having finite <a href="/Area.html">area</a>, the probability is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="126" height="45" alt=" P(R)=1-(4A^__R)/(A(R)), " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/SylvestersFour-PointProblem/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="A^__R" /> is the expected area of a triangle over region <img src="/images/equations/SylvestersFour-PointProblem/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="R" /> and <img src="/images/equations/SylvestersFour-PointProblem/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="A(R)" /> is the area of region <img src="/images/equations/SylvestersFour-PointProblem/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="R" /> (Efron 1965). Note that <img src="/images/equations/SylvestersFour-PointProblem/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="A^__R" /> is simply the value computed for an appropriate region, e.g., <a href="/DiskTrianglePicking.html">disk triangle picking</a>, <a href="/TriangleTrianglePicking.html">triangle triangle picking</a>, <a href="/SquareTrianglePicking.html">square triangle picking</a>, etc., where <img src="/images/equations/SylvestersFour-PointProblem/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="21" height="21" alt="A_R" /> can be computed exactly for <a href="/PolygonTrianglePicking.html">polygon triangle picking</a> using Alikoski's formula. </p> <p> <img src="/images/equations/SylvestersFour-PointProblem/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="P(R)" /> can range between </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="147" height="42" alt=" 2/3<=q(R)<=1-(35)/(12pi^2) " /></td><td align="right" width="3"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> (<img src="/images/equations/SylvestersFour-PointProblem/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="190" height="21" alt="0.66666<=q(R)<=0.70448" />) depending on the shape of the region, as first proved by Blaschke (Blaschke 1923, Peyerimhoff 1997). The following table gives the probabilities for various simple plane regions (Kendall and Moran 1963; Pfiefer 1989; Croft <i>et al. </i>1991, pp. 54-55; Peyerimhoff 1997). </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="R" /></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="P(R)" /></td><td align="left">approx.</td></tr><tr style=""><td align="left"><a href="/Triangle.html">triangle</a></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="27" alt="2/3" /></td><td align="left">0.66667</td></tr><tr style=""><td align="left"><a href="/Square.html">square</a></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="27" alt="(25)/(36)" /></td><td align="left">0.69444</td></tr><tr style=""><td align="left"><a href="/Pentagon.html">pentagon</a></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="99" height="28" alt="2/(45)(18-sqrt(5))" /></td><td align="left">0.70062</td></tr><tr style=""><td align="left"><a href="/Hexagon.html">hexagon</a></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="25" height="27" alt="(683)/(972)" /></td><td align="left">0.70267</td></tr><tr style=""><td align="left"><a href="/Ellipse.html">ellipse</a>, <a href="/Disk.html">disk</a></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="62" height="29" alt="1-(35)/(12pi^2)" /></td><td align="left">0.70448</td></tr> </table> </div> <p> Sylvester's problem can be generalized to ask for the probability that the <a href="/ConvexHull.html">convex hull</a> of <img src="/images/equations/SylvestersFour-PointProblem/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n+2" /> randomly chosen points in the <a href="/UnitBall.html">unit ball</a> <img src="/images/equations/SylvestersFour-PointProblem/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="B^n" /> has <img src="/images/equations/SylvestersFour-PointProblem/Inline20.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="n+1" /> vertices. The solution is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="189" height="110" alt=" P_n=((n+2)(n+1; 1/2(n+1))^(n+1))/(2^n((n+1)^2; 1/2(n+1)^2)) " /></td><td align="right" width="3"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr> </table> </div> <p> (Kingman 1969, Groemer 1973, Peyerimhoff 1997), which is the maximum possible for any bounded <a href="/ConvexDomain.html">convex domain</a> <img src="/images/equations/SylvestersFour-PointProblem/Inline21.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="52" height="21" alt="K in R^n" />. The first few values are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/SylvestersFour-PointProblem/Inline22.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="P_1" /></td><td align="center" width="14"><img src="/images/equations/SylvestersFour-PointProblem/Inline23.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline24.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="9" height="20" alt="1" /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/SylvestersFour-PointProblem/Inline25.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="P_2" /></td><td align="center" width="14"><img src="/images/equations/SylvestersFour-PointProblem/Inline26.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline27.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="41" height="42" alt="(35)/(12pi^2)" /></td><td align="right" width="10"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/SylvestersFour-PointProblem/Inline28.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="P_3" /></td><td align="center" width="14"><img src="/images/equations/SylvestersFour-PointProblem/Inline29.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline30.svg" class="displayformula" style="max-width:100%;max-height:100%;" width="24" height="26" border="0" alt="9/(143)" /></td><td align="right" width="10"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/SylvestersFour-PointProblem/Inline31.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="P_4" /></td><td align="center" width="14"><img src="/images/equations/SylvestersFour-PointProblem/Inline32.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline33.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="78" height="42" alt="(676039)/(648000pi^4)" /></td><td align="right" width="10"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/SylvestersFour-PointProblem/Inline34.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="18" height="20" alt="P_5" /></td><td align="center" width="14"><img src="/images/equations/SylvestersFour-PointProblem/Inline35.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/Inline36.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="79" height="39" alt="(20000)/(12964479)" /></td><td align="right" width="10"> <div id="eqn8" class="eqnum"> (8) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A051050">A051050</a> and <a href="http://oeis.org/A051051">A051051</a>). </p> <p> Another generalization asks the probability that <img src="/images/equations/SylvestersFour-PointProblem/Inline37.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> randomly chosen points in a fixed bounded <a href="/ConvexDomain.html">convex domain</a> <img src="/images/equations/SylvestersFour-PointProblem/Inline38.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="52" height="21" alt="K subset R^2" /> are the vertices of a convex <img src="/images/equations/SylvestersFour-PointProblem/Inline39.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />-gon. The solution is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/NumberedEquation4.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="154" height="44" alt=" P_n=(2^n(3n-3)!)/([(n-1)!]^3(2n)!) " /></td><td align="right" width="3"> <div id="eqn9" class="eqnum"> (9) </div> </td></tr> </table> </div> <p> for a triangular domain, which has first few values 1, 1, 1, 2/3, 11/36, 91/900, 17/675, ... (OEIS <a href="http://oeis.org/A004677">A004677</a> and <a href="http://oeis.org/A004824">A004824</a>), and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/SylvestersFour-PointProblem/NumberedEquation5.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="147" height="44" alt=" P_n=[1/(n!)(2n-2; n-1)]^2 " /></td><td align="right" width="3"> <div id="eqn10" class="eqnum"> (10) </div> </td></tr> </table> </div> <p> for a parallelogram domain, which has first few values 1, 1, 1, 25/36, 49/144, 121/3600, ... (OEIS <a href="http://oeis.org/A004936">A004936</a> and <a href="http://oeis.org/A005017">A005017</a>; Valtr 1996, Peyerimhoff 1997). </p> <p> Sylvester's four-point problem has an unexpected connection with the <a href="/RectilinearCrossingNumber.html">rectilinear crossing number</a> of graphs (Finch 2003). </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/DiskTrianglePicking.html">Disk Triangle Picking</a>, <a href="/HexagonTrianglePicking.html">Hexagon Triangle Picking</a>, <a href="/PolygonTrianglePicking.html">Polygon Triangle Picking</a>, <a href="/RectilinearCrossingNumber.html">Rectilinear Crossing Number</a>, <a href="/SquareTrianglePicking.html">Square Triangle Picking</a>, <a href="/TriangleTrianglePicking.html">Triangle Triangle Picking</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=beta+distribution">beta distribution</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=det+%7B%7Ba%2C+b%2C+c%7D%2C+%7Bd%2C+e%2C+f%7D%2C+%7Bg%2C+h%2C+j%7D%7D">det {{a, b, c}, {d, e, f}, {g, h, j}}</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=NC+vs+NLINSPACE">NC vs NLINSPACE</a></li> </ul> </div> </div>   <h2>References</h2><cite>Alikoski, H. A. "Über das Sylvestersche Vierpunktproblem." <i>Ann. Acad. Sci. Fenn.</i> <b>51</b>, No. 7, 1-10, 1939.</cite><cite>Blaschke, W. "Über affine Geometrie XI: Lösung des 'Vierpunktproblems' von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten." <i>Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Phys. Kl.</i> <b>69</b>, 436-453, 1917.</cite><cite>Blaschke, W. §24-25 in <i>Vorlesungen über Differentialgeometrie, II. Affine Differentialgeometrie.</i> Berlin: Springer-Verlag, 1923.</cite><cite>Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Random Polygons and Polyhedra." §B5 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387975063/ref=nosim/ericstreasuretro">Unsolved Problems in Geometry.</a></i> New York: Springer-Verlag, pp. 54-57, 1991.</cite><cite>Crofton, M. W. "Probability." <i>Encyclopedia Britannica, Vol. 19, 9th ed.</i> pp. 768-788, 1885.</cite><cite>Efron, B. "The Convex Hull of a Random Set of Points." <i>Biometrika</i> <b>52</b>, 331-343, 1965.</cite><cite>Finch, S. R. "Rectilinear Crossing Constant." §8.18 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521818052/ref=nosim/ericstreasuretro">Mathematical Constants.</a></i> Cambridge, England: Cambridge University Press, pp. 532-534, 2003.</cite><cite>Groemer, H. "On Some Mean Values Associated with a Randomly Selected Simlpex in a Convex Set." <i>Pacific J. Math.</i> <b>45</b>, 525-533, 1973.</cite><cite>Kendall, M. G. and Moran, P. A. P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/B0006AYBMM/ref=nosim/ericstreasuretro">Geometrical Probability.</a></i> New York: Hafner, 1963.</cite><cite>Kingman, J. F. C. "Random Secants of a Convex Body." <i>J. Appl. Prob.</i> <b>6</b>, 660-672, 1969.</cite><cite>Klee, V. "What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body." <i>Amer. Math. Monthly</i> <b>76</b>, 286-288, 1969.</cite><cite>Peyerimhoff, N. "Areas and Intersections in Convex Domains." <i>Amer. Math. Monthly</i> <b>104</b>, 697-704, 1997.</cite><cite>Pfiefer, R. E. "The Historical Development of J. J. Sylvester's Four Point Problem." <i>Math. Mag.</i> <b>62</b>, 309-317, 1989.</cite><cite>Rottenberg, R. R. "On Finite Sets of Points in <img src="/images/equations/SylvestersFour-PointProblem/Inline40.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="21" alt="P^3" />." <i>Israel J. Math.</i> <b>10</b>, 160-171, 1971.</cite><cite>Santaló, L. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201135000/ref=nosim/ericstreasuretro">Integral Geometry and Geometric Probability.</a></i> Reading, MA: Addison-Wesley, 1976.</cite><cite>Scheinerman, E. and Wilf, H. S. "The Rectilinear Crossing Number of a Complete Graph and Sylvester's 'Four Point' Problem of Geometric Probability." <i>Amer. Math. Monthly</i> <b>101</b>, 939-943, 1994.</cite><cite>Sloane, N. J. A. Sequences <a href="http://oeis.org/A004677">A004677</a>, <a href="http://oeis.org/A004824">A004824</a>, <a href="http://oeis.org/A004936">A004936</a>, <a href="http://oeis.org/A005017">A005017</a>, <a href="http://oeis.org/A051050">A051050</a>, and <a href="http://oeis.org/A051051">A051051</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>Solomon, H. "Crofton's Theorem and Sylvester's Problem in Two and Three Dimensions." Ch. 5 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0898710251/ref=nosim/ericstreasuretro">Geometric Probability.</a></i> Philadelphia, PA: SIAM, pp. 97-125, 1978.</cite><cite>Sylvester, J. J. "Question 1491." <i>The Educational Times (London).</i> April 1864.</cite><cite>Sylvester, J. J. "On a Special Class of Questions on the Theory of Probabilities." <i>Birmingham British Assoc. Rept.</i>, pp. 8-9, 1865.</cite><cite>Valtr, P. "Probability that <img src="/images/equations/SylvestersFour-PointProblem/Inline41.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> Random Points are in a Convex Position." <i>Discrete Comput. Geom.</i> <b>13</b>, 637-643, 1995.</cite><cite>Valtr, P. "The Probability that <img src="/images/equations/SylvestersFour-PointProblem/Inline42.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> Random Points in a Triangle are in Convex Position." <i>Combinatorica</i> <b>16</b>, 567-573, 1996.</cite><cite>Weil, W. and Wieacker, J. "Stochastic Geometry." Ch. 5.2 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0444895981/ref=nosim/ericstreasuretro">Handbook of Convex Geometry</a></i> (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 1391-1438, 1993.</cite><cite>Wilf, H. "On Crossing Numbers, and Some Unsolved Problems." In <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521584728/ref=nosim/ericstreasuretro">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</a></i> (Ed. B. Bollobás and A. Thomason). Cambridge, England: Cambridge University Press, pp. 557-562, 1997.</cite><cite>Woolhouse, W. S. B. "Some Additional Observations on the Four-Point Problem." <i>Mathematical Questions, with Their Solutions, from the Educational Times, Vol. 7.</i> London: F. Hodgson and Son, p. 81, 1867.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/sylvester%E2%80%99s_four-point_problem/80/uy/tb/" title="Sylvester's Four-Point Problem" target="_blank">Sylvester's Four-Point Problem</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Sylvester's Four-Point Problem." 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