Given a simplex of unit content in Euclidean -space,
pick
points uniformly and independently at random, and denote the expected content
of their convex hull by . Exact values are known only for and 2.
(1)
(2)
(Buchta 1984, 1986), giving the first few values 0, 1/3, 1/2, 3/5, 2/3, 5/7, ...
(OEIS A026741 and A026741).
(3)
(4)
where
is a harmonic number (Buchta 1984, 1986), giving
the first few values 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, ... (OEIS
A093762 and A093763).
Not much is known about , although
(5)
(Buchta 1983, 1986) and
(6)
(Buchta 1986).
Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex
for arbitrary .
Buchta, C. "Über die konvexe Hülle von Zufallspunkten in Eibereichen." Elem. Math.38, 153-156, 1983.Buchta,
C. "Zufallspolygone in konvexen Vielecken." J. reine angew. Math.347,
212-220, 1984.Buchta, C. "A Note on the Volume of a Random Polytope
in a Tetrahedron." Ill. J. Math.30, 653-659, 1986.Buchta,
C. and Reitzner, M. "What Is the Expected Volume of a Tetrahedron whose Vertices
are Chosen at Random from a Given Tetrahedron." Anz. Österreich. Akad.
Wiss. Math.-Natur. Kl.129, 63-68, 1992.Buchta, C. and Reitzner,
M. "The Convex Hull of Random Points in a Tetrahedron: Solution of Blaschke's
Problem and More General Results." J. reine angew. Math.536,
1-29, 2001.Klee, V. "What is the Expected Volume of a Simplex whose
Vertices are Chosen at Random from a Given Convex Body." Amer. Math. Monthly76,
286-288, 1969.Sloane, N. J. A. Sequences A026741,
A093762, and A093763
in "The On-Line Encyclopedia of Integer Sequences."
-space,
pick 
points uniformly and independently at random, and denote the expected content
of their convex hull by 
. Exact values are known only for 
and 2.
is a harmonic number (Buchta 1984, 1986), giving
the first few values 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, ... (OEIS
A093762 and A093763).
, although
points chosen at random in a three-dimensional simplex
for arbitrary 
.
