On the distribution of a random triangle

Abstract

In this paper we find the distribution of the area X of a randomly chosen triangle within a given triangle T. An extension of the classical Crofton technique is used in setting up a sequence of differential equations to obtain the desired distribution.

Central limit theorems for random polytopes

“…Efron's identity [11] states for a Poisson point process that IEf 0 ( n ) = n(V (K) − IEV ( n )), and thus the results concerning IEV ( n ) can be used to determine the expected number of vertices f 0 ( n ). Corresponding to (1),…”

“…Starting with, the only cases where a distribution function is known explicitly concerns V (P 3 ) in the planar case, and if K is a triangle (Alagar [1]), a circular disc or a parallelogram (Henze [15]); for arbitrary n, or in higher dimensions it is hopeless to expect explicit formulae.…”

“…Thus the principal curvatures k i of ∂Q(y) at y are given by the principal curvatures k i (y) of ∂K at y, and κ = κ(y) is the Gaussian curvature at the boundary point y. We denote by 1 2 Q(y), respectively 2Q(y), the paraboloid touching ∂K at y having twice, respectively half, the principal curvatures of ∂K at y. Since K ∈ K 2 + there is a constant h 0 = h 0 (K) such that for all h ≤ h 0 and for all y ∈ ∂K …”

An Identity Relating Moments of Functionals of Convex Hulls

“…The distribution function (implying all moments) of V n has apparently only been derived in the case d = 2 and n = 3: by Alagar [2] if K is a triangle and by Henze [20] if K is a circular disk or a parallelogram. All moments of V n in the case d = 2 and n = 3 had been obtained before by Miles [31] for the circular disk and by Reed [33] for the triangle and the parallelogram.…”

“…Presumably, Corollaries 2 and 3 can be improved and extended as follows: If Conjecture 1 turns out to be true, α(d) and β(d) are determined in the case d = 2 by the constants c 1 and c 2 mentioned in the Introduction and in Section 5.3: α(2) = π 1/3 c 2 , β(2) = π 1/3 ( 1 3 c 1 + c 2 ) = 1 3 η(2) + α (2). Corollary 1 and (3.6) yield for arbitrary d the relation …”