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{"Title": "Probability that $n$ uniform random points of a convex body are in convex position", "Definition": "The convex-position probability $p_K(n)$ is the probability that $n$ independent points sampled uniformly by area in a planar convex body $K$ are all vertices of their convex hull CITE{Marckert}.", "Parameters": {"region": {"type": "Symbolic", "title": "affine class of the planar convex body", "display": "$K$", "constraints": "A compact convex subset of the plane with nonempty interior, considered up to invertible affine transformation.", "values": {"equilateral-triangle": "triangle", "unit-square": "parallelogram", "unit-disk": "ellipse"}}, "n": {"type": "Z", "title": "number of sampled points", "display": "$n$", "constraints": "$n\\geq 3$."}}, "Tags": ["probability theory", "elementary geometry"], "Data properties": {"type": "R", "rigour": "proven", "complete": "no", "complete-note": "it holds every $n$ from $3$ through $8$ for the affine classes of triangles, parallelograms and ellipses; this common range includes the classical four-point problem and all five nontrivial disk probabilities explicitly displayed in the cited preprint, without gaps between the three classes", "sources": ["CITE{ValtrSquare}", "CITE{ValtrTriangle}", "CITE{Marckert}"], "rigour details": "Thirteen entries are exact Sage rationals. The five nontrivial ellipse probabilities use Arb ball arithmetic with 80 guard bits beyond 100 significant digits. Their maximum measured ball radius is less than $2\\times10^{-123}$, and the least relative accuracy exceeds 120 decimal digits. Their exact expressions are transcribed from equation (7) of CITE{DiskPreprint}, which gives the complementary probabilities $1-p_D(n)$ for a disk $D$. The parallelogram formula also appears in the author's abstract CITE{ValtrLecture}.\n\nThe rational recurrence computations agree with independent factorial evaluations throughout the range. For parallelograms, direct enumeration of Dyck paths gives the same Catalan counts, also checked against the stored HREF{Carlitz-Riordan_q-Catalan_numbers_C_n_q}[Carlitz-Riordan polynomials] at $q=1$. Disk segment convolution integrals from Theorem 1 of the preprint are evaluated independently at 130 and 170 decimal digits for $3\\leq n\\leq7$, with the known value at $n=3$ serving as a normalization control. At $n=8$, a second high-precision evaluation rearranges the same published closed form; it is not an independent derivation.\n\nA separate line-coordinate integral of the expected area of a random triangle verifies the disk four-point value. The change of variables is first checked against the known product of two disk areas. Uniform-point simulations use 500000 samples for each of the three regions and test all six sample sizes, together with sampler moments, known geometric controls and affine transformations. Every estimate differs from its predicted probability by less than two standard errors. The simulations and numerical quadratures are diagnostic checks. The exact arithmetic and Arb enclosures of the published formulas justify the stored precision. The attached checks also test the entries read back from the database."}, "Numbers": {"equilateral-triangle": {"3": {"number": "1", "comment": "Three independent uniform area points are noncollinear with probability $1$."}, "4": {"number": "2/3", "comment": "This is Sylvester's four-point probability."}, "5": "11/36", "6": "91/900", "7": "17/675", "8": "323/66150"}, "unit-square": {"3": {"number": "1", "comment": "Three independent uniform area points are noncollinear with probability $1$."}, "4": {"number": "25/36", "comment": "This is Sylvester's four-point probability."}, "5": "49/144", "6": "49/400", "7": "121/3600", "8": "20449/2822400"}, "unit-disk": {"3": {"number": "1", "comment": "Three independent uniform area points are noncollinear with probability $1$."}, "4": {"number": "0.7044798810431814999553515656382943865289535738726142325336403236419950638601466298589729510502696403", "comment": "This is Sylvester's four-point probability, $1-35/(12\\pi^2)$ CITE{Marckert}."}, "5": {"number": "0.3561883122726454106170159108548556277952202859367667208768592765057749605524623007641910719309445735", "comment": "The exact value is $1-305/(48\\pi^2)$ CITE{Marckert}."}, "6": {"number": "0.1343093863571099388567519390562815222062074839436436122362857629960799547838824432027786620320200015", "comment": "The exact value is $1-(146400\\pi^2-473473)/(11520\\pi^4)$ CITE{Marckert}."}, "7": {"number": "0.03909056230070790319009578457274220307412526130314250142005362829091538567473484718778033253057964058", "comment": "The exact value is $1-(512400\\pi^2-2900611)/(23040\\pi^4)$ CITE{Marckert}."}, "8": {"number": "0.009034262164855343163545962523547630210849646555099686962509365970595457222086937730075448400680513119", "comment": "The exact value is $1-(62664108221+1721664000\\pi^4-18670881600\\pi^2)/(48384000\\pi^6)$ CITE{Marckert}."}}}, "Comments": {"affine": "The parameter $K$ is the body argument of the probability $p_K(n)$. The region labels represent affine classes. The triangle row applies to every nondegenerate triangle, the parallelogram row to every nondegenerate parallelogram, and the ellipse row to every filled nondegenerate ellipse. Uniformity means normalized two-dimensional area measure, including the interior.", "sampling": "The points are sampled independently. There is no imposed cyclic order, no fixed boundary point, and no conditioning on the convex hull. Collinear triples and repeated points have probability zero.", "small": "The case $n=4$ is Sylvester's four-point problem CITE{Marckert}. At $n=3$ the probability is $1$ for every planar convex body with nonempty interior.", "exactness": "The triangle and parallelogram entries are exact rational numbers. The nontrivial ellipse entries are decimal enclosures of the exact expressions in their comments."}, "Formulas": {"affine": "$p_{AK+b}(n)=p_K(n)$ for every invertible real $2\\times2$ matrix $A$ and every $b\\in\\mathbb R^2$. An invertible affine map preserves vertices and sends normalized area measure to normalized area measure.", "triangle": "For a triangle $T$, $p_T(n)=\\dfrac{2^n(3n-3)!}{((n-1)!)^3(2n)!}$ for $n\\geq3$ CITE{ValtrTriangle}.", "square": "For a parallelogram $S$, $p_S(n)=\\left(\\dfrac{\\binom{2n-2}{n-1}}{n!}\\right)^2$ for $n\\geq3$ CITE{ValtrSquare}.", "catalan": "If $C_m(q)$ is the HREF{Carlitz-Riordan_q-Catalan_numbers_C_n_q}[Carlitz-Riordan $q$-Catalan polynomial], then $p_S(n)=\\left(C_{n-1}(1)/(n-1)!\\right)^2$. The specialization $C_m(1)$ counts Dyck paths of semilength $m$.", "triangle-recurrence": "$\\dfrac{p_T(n+1)}{p_T(n)}=\\dfrac{2(3n)(3n-1)(3n-2)}{n^3(2n+2)(2n+1)}$ for $n\\geq3$, with $p_T(3)=1$.", "square-recurrence": "$\\dfrac{p_S(n+1)}{p_S(n)}=\\left(\\dfrac{2(2n-1)}{n(n+1)}\\right)^2$ for $n\\geq3$, with $p_S(3)=1$.", "area": "$p_K(4)=1-4\\,\\mathbf E[\\operatorname{Area}(\\operatorname{conv}\\{X_1,X_2,X_3\\})]/\\operatorname{Area}(K)$. Outside a null set, a nonconvex configuration of four points has exactly one point inside the triangle formed by the other three.", "monotone": "$p_K(n+1)\\leq p_K(n)$ for $n\\geq3$, since every subset of a set in convex position is itself in convex position."}, "References": {"ValtrSquare": {"bib": "P. Valtr, Probability that n random points are in convex position, Discrete & Computational Geometry 13 (1995), 637\u2013643.", "doi": "10.1007/BF02574070"}, "ValtrTriangle": {"bib": "P. Valtr, The probability that n random points in a triangle are in convex position, Combinatorica 16 (1996), 567\u2013573.", "doi": "10.1007/BF01271274"}, "Marckert": {"bib": "J.-F. Marckert, The probability that n random points in a disk are in convex position, Brazilian Journal of Probability and Statistics 31 (2017), 320\u2013337.", "doi": "10.1214/16-BJPS315", "arxiv": "1402.3512"}}, "Links": {"DiskPreprint": {"title": "Marckert: explicit disk formulas and the segment-integral recurrence (preprint)", "url": "https://arxiv.org/abs/1402.3512"}, "ValtrLecture": {"title": "Valtr: On the probability that n random points are in convex position, DIMACS seminar abstract", "url": "https://archive.dimacs.rutgers.edu/Events/1996/Titles/1996/Valtr.html"}}, "Similar tables": [{"table": "HREF{Carlitz-Riordan_q-Catalan_numbers_C_n_q}[Carlitz-Riordan q-Catalan numbers]", "relation": "Their values at $q=1$ give the Catalan numbers in the parallelogram probability."}], "Display properties": {"number-header": "$p_K(n)$", "group parameters": [["region"], ["n"]]}, "Keywords": ["convex position", "Sylvester four-point problem", "random convex polygon", "geometric probability", "Valtr probability", "Marckert disk probability", "triangle", "parallelogram", "square", "disk", "ellipse"], "Programs": {"polygon": {"language": "Sage", "code": "# Exact probabilities for n uniform area points in a triangle or parallelogram.\nimport numberdb.sage as numberdb\nfrom sage.rings.rational_field import QQ\nfrom math import comb, factorial\nn = 4\np_triangle = QQ(2**n * factorial(3*n-3)) / QQ(factorial(n-1)**3 * factorial(2*n))\np_parallelogram = (QQ(comb(2*n-2, n-1)) / QQ(factorial(n)))**2\nprint(p_triangle, p_parallelogram)\n"}, "disk-four": {"language": "Sage", "code": "# Sylvester's probability for four uniform area points in any ellipse.\nimport numberdb.sage as numberdb\nfrom sage.rings.real_arb import RealBallField\nR = RealBallField(413)\nprint(1 - R(35)/(12*R.pi()**2))\n"}}}