{"Title": "Expected number of vertices of the convex hull of $n$ uniform random points", "Definition": "For independent points $X_1,\\ldots,X_n$ uniformly distributed by volume throughout a convex body $K$, let $N_n(K)$ be the number of vertices of their convex hull. This table gives the single quantity $e_K(n)=\\mathbf E[N_n(K)]$ as a function of the convex body $K$ and the sample size $n$ CITE{Efron}.", "Parameters": {"region": {"type": "Symbolic", "title": "affine class of the convex body", "display": "$K$", "constraints": "A compact convex subset of $\\mathbb R^d$ with nonempty interior, considered up to invertible affine transformations.", "values": {"equilateral-triangle": "triangle", "unit-square": "parallelogram", "unit-disk": "ellipse"}}, "n": {"type": "Z", "title": "number of sampled points", "display": "$n$", "constraints": "$n\\geq1$."}}, "Tags": ["probability theory", "elementary geometry"], "Data properties": {"type": "R", "rigour": "proven", "complete": "no", "complete-note": "it holds every $n$ from $4$ through $20$ for the affine classes of triangles, parallelograms and filled ellipses, covering the first nontrivial four-point case and small finite-sample hulls in a common range; other convex bodies and higher dimensions are not included", "sources": ["CITE{Buchta}", "CITE{Efron}"], "rigour details": "The 34 triangle and parallelogram values use exact rational arithmetic. The 17 ellipse values use Arb's certified integration of an entire trigonometric integrand, with 80 guard bits beyond 100 significant digits. The maximum measured absolute radius is less than $1.34\\times10^{-121}$. Every enclosure is required to be finite and to have relative radius below $10^{-110}$.\n\nIndependent exact integration of hull-edge probabilities reproduces every rational entry. For triangles this integrates over the three pairs of boundary edges. For squares it separates the four adjacent and two opposite pairs. The controls are the known three-point expectation $3$ and the four-point expectations $11/3$ and $133/36$.\n\nA separate signed-area integral from printed page 15 of CITE{EfronReport}, combined with Efron's identity, checks each ellipse value at 130 and 170 decimal digits. The mean area fraction $35/(48\\pi^2)$ for three disk points is tested first as a normalization control. The two quadratures agree within $10^{-125}$, and they agree with the Arb values within $10^{-115}$. The expected-edge integral also passes its three-point control $3$ and the four-point closed form $4-35/(12\\pi^2)$. Numerical quadrature supplies agreement checks; the Arb enclosures justify the stored digits.\n\nThe attached checks compare all 51 entries read back from the database with the independent computations, test all 51 Efron relations to HREF{T470}[the hull-volume ratios], test the three four-point relations to HREF{T468}[the convex-position probabilities], and check $3<e_K(n)<n$ for every stored entry."}, "Numbers": {"equilateral-triangle": {"4": "11/3", "5": "25/6", "6": "137/30", "7": "49/10", "8": "363/70", "9": "761/140", "10": "7129/1260", "11": "7381/1260", "12": "83711/13860", "13": "86021/13860", "14": "1145993/180180", "15": "1171733/180180", "16": "1195757/180180", "17": "2436559/360360", "18": "42142223/6126120", "19": "14274301/2042040", "20": "275295799/38798760"}, "unit-square": {"4": "133/36", "5": "305/72", "6": "281/60", "7": "3647/720", "8": "217687/40320", "9": "51031/8960", "10": "1442363/241920", "11": "187627/30240", "12": "68410249/10644480", "13": "141187553/21288960", "14": "472038551/69189120", "15": "553530163/79073280", "16": "9067345117/1265172480", "17": "18531607469/2530344960", "18": "107136683491/14338621440", "19": "327447366043/43015864320", "20": "12660946977401/1634602844160"}, "unit-disk": {"4": {"number": "3.704479881043181499955351565638294386528953573872614232533640323641995063860146629858972951050269640", "comment": "The exact value is $4-35/(12\\pi^2)$, obtained from the disk case of Sylvester's four-point problem."}, "5": "4.261199702607953749888378914095735966322383934681535581334100809104987659650366574647432377625674101", "6": "4.727567555779794174307890231975216459751800405887649046835200472959323961470205249897636846539149674", "7": "5.132287486101441485468289612578181727002957649221397129253848823490177056369435363375715641197164507", "8": "5.491919496457589379714691889836316711329966522042113139617636620635170555580531819898810225145553968", "9": "5.816951568403909045437567267996425897521115979523575794359495250158153332427320622809568151999837037", "10": "6.114478823847501751007406437611507912141658009945466229727314656110694862858506404301917549344774190", "11": "6.389543105887125138742053133536292823655200102339096508988752307062832895714436190583954360684146334", "12": "6.645866482340570441298673249614638350654983888797519313967937433418786463528402786500661943116637614", "13": "6.886281710615615159278042231712855524150106874117828228748122900275151563595217969886546842874770151", "14": "7.112999077036621083201248920759196488457765697146154149682405112130292378712128255987655776506333524", "15": "7.327779317182812601438864931619620782977060146010610588783572147599284472294609612122706331588030433", "16": "7.532049871184049434569637626370679894667474231407307394245099250432850990350095283106201190372809678", "17": "7.726985501614797536094269226895721594828939163094485790185790267210775201682949778132964324528232517", "18": "7.913565697361177510349950081563984501723144856020179004509967140874343525577719086427975190054313552", "19": "8.092616493589788515127220062319525840603014993919363114803975115417458273573133268783749000487982523", "20": "8.264841553092724186426776334570296333005260333127557471772667634613102053280288624950691397277578266"}}, "Comments": {"affine": "The body $K$ is an argument of the single quantity $e_K(n)$. Each region label denotes an affine class: all nondegenerate triangles, all nondegenerate parallelograms, or all filled nondegenerate ellipses. An invertible affine map preserves vertices and carries normalized volume measure to normalized volume measure.", "sampling": "The points are independent, with no imposed cyclic order and no conditioning on their hull. They are sampled throughout the interior, rather than on the boundary. Coincident points and affine dependencies among at most $d+1$ points have probability zero.", "small": "For $1\\leq n\\leq d+1$, all sampled points are vertices with probability one, so $e_K(n)=n$. In the plane these universal cases are omitted; the first nontrivial case is $n=4$.", "exactness": "The triangle and parallelogram entries are exact rationals. The ellipse entries are decimal enclosures with 100 significant digits. The expectation is a real number although each realized hull has an integer number of vertices."}, "Formulas": {"affine": "$e_{AK+b}(n)=e_K(n)$ for every invertible real $d\\times d$ matrix $A$ and every $b\\in\\mathbb R^d$.", "efron": "$e_K(n)=n\\bigl(1-v_K(n-1)\\bigr)$ for $n\\geq2$, where $v_K(m)$ is the HREF{T470}[expected hull-volume ratio] for $m$ independent uniform points in $K$ CITE{Efron} CITE{Duality}. A specified sample point is a vertex exactly when it lies outside the convex hull of the other points.", "region-triangle": "For a triangle $T$ and $n\\geq2$, $e_T(n)=2H_{n-1}$, where $H_m=\\sum_{k=1}^m1/k$ is the $m$th HREF{Values_of_the_harmonic-number_function_at_rational_numbers}[harmonic number] CITE{Buchta} CITE{BuchtaLecture}.", "region-parallelogram": "For a parallelogram $S$ and $n\\geq2$, $e_S(n)=\\dfrac83\\left(\\sum_{k=1}^n\\dfrac{1-2^{-k}}{k}-\\dfrac{1}{n2^n}\\right)$ CITE{Buchta} CITE{BuchtaLecture}.", "region-ellipse": "For a filled ellipse $D$ and $n\\geq2$, put $F(\\theta)=(\\theta-\\sin\\theta\\cos\\theta)/\\pi$. Then $e_D(n)=\\dfrac{8n(n-1)}{3\\pi}\\int_0^\\pi\\sin^4\\theta\\,F(\\theta)^{n-2}\\,d\\theta$ CITE{Efron}. For the unit disk, $F(\\theta)$ is the area fraction to the left of the vertical line at coordinate $-\\cos\\theta$.", "sylvester": "For a planar convex body $K$, $e_K(4)=3+p_K(4)$, where $p_K(4)$ is HREF{T468}[the probability that four independent uniform points are in convex position]. The hull has four vertices in convex position and three otherwise, outside a null set.", "bounds": "For a planar convex body and $n\\geq4$, $3<e_K(n)<n$."}, "Display properties": {"number-header": "$e_K(n)$", "group parameters": [["region"], ["n"]]}, "References": {"Efron": {"bib": "B. Efron, The convex hull of a random set of points, Biometrika 52 (1965), 331\u2013343.", "doi": "10.1093/biomet/52.3-4.331"}, "Buchta": {"bib": "C. Buchta, Zufallspolygone in konvexen Vielecken, Journal f\u00fcr die reine und angewandte Mathematik 347 (1984), 212\u2013220.", "doi": "10.1515/crll.1984.347.212"}, "Duality": {"bib": "C. Buchta, The duality of the volumes and the numbers of vertices of random polytopes, Discrete & Computational Geometry 70 (2023), 951\u2013959.", "doi": "10.1007/s00454-023-00482-4"}}, "Links": {"BuchtaLecture": {"title": "Buchta: The convex hull of randomly chosen points, Rouen lecture (2012), slide 5", "url": "https://gdr-geostoch.math.cnrs.fr/workshop_Rouen/SlidesRouen2012/Friday/Vortrag_Buchta.pdf"}, "EfronReport": {"title": "Efron: The convex hull of a random set of points, Stanford technical report SOL-ONR-103", "url": "https://purl.stanford.edu/fy983bv8306"}}, "Similar tables": [{"table": "HREF{T470}[Expected hull-volume ratios]", "relation": "Efron's identity converts the expected volume fraction for $n-1$ points into the expected vertex count for $n$ points."}, {"table": "HREF{T468}[Convex-position probabilities]", "relation": "The four-point probability is $e_K(4)-3$ in the plane."}, {"table": "HREF{Values_of_the_harmonic-number_function_at_rational_numbers}[Harmonic numbers]", "relation": "For a triangle, the expected count is twice $H_{n-1}$."}], "Keywords": ["expected convex hull vertices", "expected hull complexity", "random polygon", "random polytope", "geometric probability", "Efron identity", "Buchta", "triangle", "parallelogram", "square", "disk", "ellipse"], "Programs": {"polygon": {"language": "Sage", "code": "import numberdb.sage as numberdb\nfrom sage.rings.rational_field import QQ\n\ndef triangle_vertices(n):\n    return 2*sum(QQ(1)/k for k in range(1,n))\n\ndef parallelogram_vertices(n):\n    s=sum((1-QQ(1)/2**k)/k for k in range(1,n+1))\n    return QQ(8)/3*(s-QQ(1)/(n*2**n))\n\nprint(triangle_vertices(4), parallelogram_vertices(4))\n"}, "ellipse": {"language": "Sage", "code": "import numberdb.sage as numberdb\nfrom sage.rings.complex_arb import ComplexBallField\n\ndef ellipse_vertices(n, digits=100):\n    C=ComplexBallField(numberdb.bits(digits,losing=80))\n    pi=C.pi()\n    def integrand(u,analytic):\n        t=pi*u\n        return t.sin()**4*(u-t.sin()*t.cos()/pi)**(n-2)\n    return (C(8*n*(n-1))/3*C.integral(integrand,0,1)).real()\n\nprint(ellipse_vertices(4))\n"}}}