(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 A002548).
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 A002548,
A026741, and A093762
in "The On-Line Encyclopedia of Integer Sequences."
points uniformly and independently
at random in a simplex of unit content
in Euclidean space of dimension 
. For positive
integers 
and 
,
denote the expectation value of the 
-dimensional content of their convex hull by 
, which is 0 for 
.
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 A002548).
,
points chosen at random in a three-dimensional simplex
for arbitrary 
.
