lex Simplex Picking
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.
(Buchta 1984, 1986), giving the first few values 0, 1/3, 1/2, 3/5, 2/3, 5/7, ... (OEIS A026741 and A026741).
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 
.
REFERENCES:
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. Monthly 76, 286-288, 1969.
Sloane, N. J. A. Sequences A026741, A093762, and A093763 in "The On-Line Encyclopedia of Integer Sequences."
