OFFSET
0,2
COMMENTS
Mean volume of a tetrahedron formed by four random points in a unit ball.
Equals (4*Pi/15) times the probability (9/143) that 5 points independently and uniformly chosen in a ball are the vertices of a re-entrant (concave) polyhedron, i.e., one of the points falls within the tetrahedron formed by the other 4 points. It was calculated by the Czech physicist and mathematician Bohuslav Hostinský (1884 - 1951) in 1925. - Amiram Eldar, Aug 25 2020
REFERENCES
Bohuslav Hostinský, Sur les probabilités géométriques, Brno: Publications de la Faculté des sciences de l'Université Masaryk, 1925.
LINKS
Fernando Affentranger, The expected volume of a random polytope in a ball, Journal of Microscopy, Vol. 151, No. 3 (1988), pp. 277-287.
Herbert Solomon, Geometric Probability, Philadelphia, PA: SIAM, 1978, p. 124.
Eric Weisstein's World of Mathematics, Ball Tetrahedron Picking.
EXAMPLE
0.0527260305...
MATHEMATICA
RealDigits[12*Pi/715, 10, 100][[1]] (* Amiram Eldar, Aug 25 2020 *)
PROG
(PARI) 12*Pi/715 \\ Charles R Greathouse IV, Sep 30 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Apr 02 2004
STATUS
approved