OFFSET
0,3
COMMENTS
Average volume of a tetrahedron picked at random in a tetrahedron with unit volume.
Buchta & Reitzner announced this result in 1992, and Mannion (independently) proved it in 1994. Buchta & Reitzner proved a more general result in 2001. - Charles R Greathouse IV, Sep 04 2015
Klee (1969) conjectured that the average volume is 1/60 and stated that according to Monte Carlo experiments 1/57 is the integer-reciprocal closest to this value. - Amiram Eldar, Apr 09 2022
LINKS
Christian Buchta and Matthias Reitzner, What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron, Österreichische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, Vol. 129 (1992), pp. 63-68.
Christian Buchta and Matthias Reitzner, The convex hull of random points in a tetrahedron: Solution of Blaschke's problem and more general results, J. reine angew. Math., Vol. 536 (2001), pp. 1-29.
Victor Klee, What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body?, The American Mathematical Monthly, Vol. 76, No. 3 (1969), pp. 286-288.
Eric Weisstein's World of Mathematics, Tetrahedron Tetrahedron Picking.
EXAMPLE
0.0173982392...
MATHEMATICA
Join[{0}, RealDigits[13/720 - Pi^2/15015, 10, 100][[1]]] (* Amiram Eldar, Apr 09 2022 *)
PROG
(PARI) 13/720 - Pi^2/15015 \\ Charles R Greathouse IV, Sep 04 2015
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Mar 30 2004
EXTENSIONS
Added initial 0 to make offset correct. - N. J. A. Sloane, Feb 08 2015
STATUS
approved