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Decimal expansion of a constant appearing in the expression of the asymptotic expected volume V(d) of the convex hull of uniformly selected n(d) points in the interior of a d-dimensional unit cube.
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%I #7 Jun 02 2017 12:28:11

%S 2,1,3,9,6,9,0,9,4,7,4,1,2,8,5,9,8,6,0,5,0,5,3,0,2,2,6,3,8,5,2,3,5,2,

%T 4,4,4,3,2,3,1,4,6,9,5,6,0,5,5,1,2,9,3,8,5,8,2,4,9,8,0,0,0,7,6,0,1,1,

%U 1,5,5,2,1,8,3,2,5,1,3,3,3,2,3,8,9,6,9,7,2,7,1,2,4,4,0,0,5,2,3,8,4,3,2

%N Decimal expansion of a constant appearing in the expression of the asymptotic expected volume V(d) of the convex hull of uniformly selected n(d) points in the interior of a d-dimensional unit cube.

%H Steven R. Finch, <a href="/A249455/a249455.pdf">Convex Lattice Polygons</a>, December 18, 2003. [Cached copy, with permission of the author]

%F k = exp(log(2*Pi) - gamma - 1/2).

%F Lim_{d -> infinity} V(d) =

%F 0 if n(d) <= (k - epsilon)^d

%F 1 if n(d) >= (k + epsilon)^d

%e 2.139690947412859860505302263852352444323146956...

%t k = Exp[Log[2*Pi] - EulerGamma - 1/2]; RealDigits[k, 10, 103] // First

%Y Cf. A249455.

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Oct 29 2014