On some mean values associated with a randomly selected simplex in a convex set

Groemer, H.

doi:10.2140/pjm.1973.45.525

Cited by 75 publications

(44 citation statements)

References 7 publications

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“…Under these assumptions, (4) has been proved by Groemer [11] and (5) by Macbeath [20]. For s = 1 the value of E(B) has been computed explicitly by Kingman [18].…”

Section: Main Theorem and Corollariesmentioning

confidence: 96%

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On the average size of polytopes in a convex set

Groemer

1982

Geom Dedicata

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“…Under these assumptions, (4) has been proved by Groemer [11] and (5) by Macbeath [20]. For s = 1 the value of E(B) has been computed explicitly by Kingman [18].…”

Section: Main Theorem and Corollariesmentioning

confidence: 96%

“…Again it can be shown that the ellipsoids (and only these) have this property (Groemer [11] ). The mean value problem is more general since/~(K) = tim ~ (/zr(K)) 1/'.…”

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confidence: 97%

On the average size of polytopes in a convex set

Groemer

1982

Geom Dedicata

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“…Ke~ ist (aueh ftir n=2) nach BRu~N [8] genau dann ein Ellipsoid, wenn fiir jede Parallelschar yon Sehnen die Mittelpunkte in einer Hyperebene liegen. Altarnativbewaise d~von findet man in [5], [6], [10], [14], [23], Verschgrfungen bei DA:~ZE~--LAUG-WITZ--LE:SZ [11] und G~vB~ [15]. Die Ergebnisse in [17], [26] lassen sieh leieht auf das Brunnsche t~esultat zuriickfiihren.…”

Section: H Nicht Parallel G Bd (K -~ G) N Bd K ) H N Bd (K -F G) (2)unclassified

“…Wegen oeintK, (14), (18), (15), (16), (20), ~*, (15), (]8), (18), (16), (15), (20) gilt K* ~-G =-cl cony (K*w G) -----(K r~ G*)* -~ (K hE)* = : ((g -~ G (E)) n E)* -----cl cony ((K -~ G (E))* w E*) : = (K -~ G (E))* + E* : (cl cony (K w G (E)))* + G = --(K* r~ (G (E))*) -~ a --(K* n E (G)) + G. …”

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confidence: 99%

�ber kennzeichnende Eigenschaften von Ellipsoiden und euklidischen R�umen III

Gruber

1974

Monatshefte f�r Mathematik

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“…In 1974 Groemer [11,12] generalized this statement to d-dimensional convex bodies C of volume one and an arbitrary number n of points. The value of this minimum is known for d + 1 points in a d-dimensional ellipsoid (Kingman [13]) and for an arbitrary number of points in dimensions d = 2 and d = 3 (cf.…”

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confidence: 99%

Random polytopes on the torus

Buchta¹,

Tichy²

1985

Proc. Amer. Math. Soc.

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ABSTRACT.The expected volume of the convex hull of n random points chosen independently and uniformly on the d-dimensional torus is determined.In the 1860's J. J. Sylvester raised the problem of determining the expected area V$(C) of the convex hull of three points chosen independently and uniformly at random from a given plane convex body C of area one. For some special plane convex bodies the problem was solved by Woolhouse [16], Crofton [8] and Deltheil In 1917 Blaschke [3,4] succeeded in proving that, among all plane convex bodies C of area one, Vs(C) attains its minimum if C is an ellipse. In 1974 Groemer [11,12] generalized this statement to d-dimensional convex bodies C of volume one and an arbitrary number n of points. The value of this minimum is known for d + 1 points in a d-dimensional ellipsoid (Kingman [13]) and for an arbitrary number of points in dimensions d = 2 and d = 3 (cf. [6]).An obvious question is to ask for the expected volume of the convex hull of random points chosen on a compact metric manifold. A subset C of a compact metric manifold is called convex (cf. Bangert [2], Walter [14] ) if for any two points x, y G C all geodesic segments (i.e. all curves of minimal length on the manifold joining x and y) are completely contained in C. The convex hull of a set is the smallest convex set (on the manifold) containing it.In the case of the d-dimensional sphere S^ = {x G Ed+1: \\x\\ = 1} (Ed+1 denotes (d+ l)-dimensional Euclidean space), the metric is defined by the minimal Euclidean length of all curves on S^ joining two points. A convex set on S^ is either contained in a hemisphere or is the sphere itself. If n points are contained

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On some mean values associated with a randomly selected simplex in a convex set

Cited by 75 publications

References 7 publications

On the average size of polytopes in a convex set

On the average size of polytopes in a convex set

�ber kennzeichnende Eigenschaften von Ellipsoiden und euklidischen R�umen III

Random polytopes on the torus

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