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Expected convex hull volume ratios, T470
Convention and scope
For independent normalized-volume samples in a full-dimensional convex body K,
v_K(n) is E[Vol(conv(X_1,...,X_n))]/Vol(K). The rows here are the planar affine
classes of triangles, parallelograms and filled ellipses. The reference
representatives are the triangle ((0,0),(1,0),(0,1)), unit square, and unit disk.
The shared family slug equilateral-triangle identifies every triangle: no
metric normalization affects this ratio. The selected common range is 3..20,
covering small finite-sample hulls, with no trivial zero-volume rows.
Sources inspected
Buchta, The convex hull of randomly chosen points, Rouen 2012 lecture,
slides 4 and 5: Efron's identity and exact triangle and square vertex means.
https://gdr-geostoch.math.cnrs.fr/workshop_Rouen/SlidesRouen2012/Friday/Vortrag_Buchta.pdf
Buchta's 1984 original is DOI 10.1515/crll.1984.347.212; its full text was not
accessible during this build. The author lecture supplies the actual formulas.
Efron, The convex hull of a random set of points, Stanford report SOL-ONR-103
(1965), especially printed p.15 (PDF page 18): the rotationally symmetric
signed-area integral. The scanned page was read visually.
https://purl.stanford.edu/fy983bv8306
Marckert, Brazilian Journal of Probability and Statistics 31 (2017), 320-337,
DOI 10.1214/16-BJPS315, author PDF introductory discussion: the five-point
disk hull ratio 175/(72*pi^2)-23023/(6912*pi^4).
https://www.labri.fr/perso/marckert/Conv_Points_Disk.pdf
Generator disk integral
Put h=-cos(theta), F(h)=(theta-sin(theta)cos(theta))/pi. F is the fraction of
the disk on the left of an oriented chord whose outward normal distance is h.
A line-coordinate change of variables for two ordered sampled points gives
E[N_m]=m(m-1)*8/(3*pi)*integral[-1,1](1-h^2)^(3/2)*F(h)^(m-2) dh.
The factor is obtained by integrating (t2-t1) over an ordered pair on a chord
of length L=2 sqrt(1-h^2): the integral is L^3/6. Integration over normal
angle gives 2*pi, and the two normalized disk area measures give pi^-2.
The m=3 control yields exactly 3 vertices. Efron's identity
v_K(n)=1-E[N_(n+1)]/(n+1) now gives
v_D(n)=1-8*n/(3*pi)*integral[0,pi] sin(theta)^4*F(-cos(theta))^(n-1)dtheta.
The generator changes theta=pi*u to exact integration limits 0 and 1.
Arb complex integration is rigorous because sine, cosine and the finite
integer powers involved are entire. Imaginary parts must contain zero,
real balls must be finite, positive, smaller than one, and narrow enough.
Independent disk check
The shoelace contribution of an ordered hull edge is det(X_i,X_j)/2. In
line coordinates det(X_i,X_j)=h*(t2-t1). The squared chord difference
integrates to L^4/12. This yields the signed-area formula
v_D(n)=4*n*(n-1)/(3*pi^2)*integral[-1,1] h*(1-h^2)^2*F(h)^(n-2)dh.
This is also the specialization of Efron's printed p.15 formula, using
conditional variance (1-h^2)/3 and marginal density 2 sqrt(1-h^2)/pi.
checks.py integrates this formula with mpmath at 135 and 175 decimal digits,
first checking n=3 against 35/(48*pi^2). This check does not certify digits;
the certification comes from the generator's Arb balls. Closed forms at
n=3,4,5 are also checked by separate 600-bit ball evaluations.
Independent polygon check
For each ordered pair of different boundary sides, parameterize its chord
endpoints A(s), B(t), with 0<s,t<1. Write D=B-A and sample points
P=A+uD, Q=A+vD, 0<u<v<1. The transformation Jacobian is
-cross(D,A')cross(D,B')*(v-u). The signed cross product is
cross(P,Q)=(v-u)*cross(A,B). The left-side area L(s,t) is computed by the
shoelace formula along the boundary path B to A. Each remaining point is
on that side with probability L/Area(K). Integrating (v-u)^2 gives 1/12;
summing all ordered pairs and taking the expected shoelace area gives
n(n-1)/(24*Area(K)^3) times the sum of the integrals over s,t of
cross(A,B)*[-cross(D,A')cross(D,B')]*(L/Area(K))^(n-2).
All integrands are rational polynomials. Each monomial c*s^a*t^b integrates
exactly to c/((a+1)(b+1)). Controls are 1/12, 11/144, 1/6 and 11/72 for
triangle/square at n=3/4. The calculation covers all 36 rational entries,
and is repeated after the map (x,y)->(2x+y+3,x+3y-2), whose determinant is 5.
No Buchta harmonic-sum code is used by this integration.
Validation and precision
The dry run computes 54 entries; 36 are exact rational numbers. The 18 disk
entries use 80 guard bits beyond 100 digits (413 working bits). Their largest
measured relative radius is 6.012246711300344e-122. See validation.json for
every independently computed disk value. The value block is about 6 KB in
the diagnostic serialization and the longest ball representation is 140
characters. All 51 adjacent monotonicity comparisons and the three
v_K(4)=2*v_K(3) checks passed.
With a key, checks.py reads back all of T470 and checks independent rational
values, disk values against stored decimal intervals, adjacent monotonicity,
and the relation v_K(3)=(1-p_K(4))/4 with the stored T468 entries.
Reproduction
Install numberdb and SageMath (or passagemath), then run:
sage -python checks.py
NUMBERDB_API_KEY=... sage -python generate.py
The latter only verifies. generate.py --fill writes numbers and source files
to the draft; it does not publish the table or request human review.
No simulation supplies numbers or error bounds in this table.