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Simplex Simplex Picking


Given a simplex of unit content in Euclidean d-space, pick n>=d+1 points uniformly and independently at random, and denote the expected content of their convex hull by V(d,n). Exact values are known only for d=1 and 2.

V(1,n)=1-2/(n+1)
(1)
=(n-1)/(n+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).

V(2,n)=1-2/(n+1)sum_(k=1)^(n)1/k
(3)
=1-(2H_n)/(n+1),
(4)

where H_n 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 V(3,n), although

 V(3,5)=5/2V(3,4)
(5)

(Buchta 1983, 1986) and

 1-V(3,n)∼3/4((lnn)^2)/n
(6)

(Buchta 1986).

Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of n points chosen at random in a three-dimensional simplex for arbitrary n.


See also

Disk Triangle Picking, Simplex, Tetrahedron Tetrahedron Picking

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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."

Referenced on Wolfram|Alpha

Simplex Simplex Picking

Cite this as:

Weisstein, Eric W. "Simplex Simplex Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SimplexSimplexPicking.html

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