TOPICS
Simplex Simplex Picking
Simplex simplex picking consists of choosing 
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 
.
 |  |  |
(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 A002548).
For 
,
 |
(5)
|
(Buchta 1983, 1986) and
 |
(6)
|
(Buchta 1986).
Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expectation value of the
volume of the convex hull
of 
points chosen at random in a three-dimensional simplex
for arbitrary 
.
See also
Disk Triangle Picking, Simplex, Tetrahedron Tetrahedron PickingExplore with Wolfram|Alpha
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 A002548, A026741, and A093762 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Simplex Simplex PickingCite this as:
Weisstein, Eric W. "Simplex Simplex Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SimplexSimplexPicking.html