A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes between Gaussian curvature and both circumference and area deficits.

We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$-th power of their Euclidean distance, with $p\gt 0$. In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$, we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

In a family, parameterized by θ, of non-negative random variables with finite, positive second moment, Taylor's law (TL) asserts that the population variance is proportional to a power of the population mean as θ varies: σ2 (θ) = a[μ(θ)]b, a > 0. TL, sometimes called fluctuation scaling, holds widely in science, probability theory, and stochastic processes. Here we report diverse examples of TL with b = 2 (equivalent to a constant coefficient of variation) arising from a difference of random variables in normed vector spaces of dimension 1 and larger. In these examples, we compute a exactly using, in some cases, a simple, new technique. These examples may prove useful in future models that involve differences of random variables, including models of the spatial distribution and migration of human populations.

We introduce the entropy of a family of planar curves in terms of the number of intersections of the family with a random line, calculate it for key examples, and discuss the entropy of a pattern of rings produced by an impulse on the surface of still water.

The convex hull of n independent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the variance of the ith intrinsic volume of a Gaussian polytope in ℝd, d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number of i-faces and for the ith intrinsic volume of a Gaussian polytope as n→∞.

Let U1,U2,… be a sequence of i.i.d. random vectors distributed uniformly in a compact plane region A of unit area. Sufficient conditions on the geometry of A are provided under which the Euclidean diameter Dn of the first n of the points converges weakly upon suitable rescaling.

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let Mn be the longest edge-length of the minimal spanning tree on these points; equivalently let Mn be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2Mn - bn converges weakly to the Gumbel (double exponential) distribution, where bn are explicit constants with bn ~ (ν - 1)log log n. We also show the same result holds if Mn is the longest edge-length for the nearest neighbour graph on the points.

Let us consider, in the Euclidean space En, a fixed n-dimensional convex body K0 of volume V0 and a system K1,…,Km of mn-dimensional convex bodies, congruent to a convex set K. Assume that the sets Ki (i = 1,…,m) have random positions, being stochastically independent and uniformly distributed on a limited domain of En and denote by Vm the volume of the convex body Km = K0 ∩ (K1 ∩ … ∩ Km). The aim of this paper is the evaluation of the second moment of the random variable Vm.

An asymptotic expression for the expected area of the union of n random rectangles is derived by Mellin transforms, where their two diagonal corners are independently and uniformly distributed over (0,1)2. The general result for d-dimensional hyper-rectangles is also stated.

Let denote a rectangular lattice in the Euclidean plane E2, generated by (a × b) rectangles. In this paper we consider the probability that a random ellipse having main axes of length 2α and 2ß, with intersects . We regard the lattice as the union of two orthogonal sets and of equidistant lines and evaluate the probability that the random ellipse intersects or . Moreover, we consider the dependence structure of the events that the ellipse intersects or . We study further the case when the main axes of the ellipse are parallel to the lines of the lattice and satisfy 2ß = min (a, b) < 2α = max (a, b). In this case, the probability of intersection is 1, and there exist almost surely two perpendicular segments in within the ellipse. We evaluate the distribution function, density, mean and variance of the length of these segments. We conclude with a generalization of this problem in three dimensions.

Consider a compact body E embedded in a convex compact body G. A point P is randomly chosen in E and three different rays to the boundary ∂G of G are generated by P. Ray is in a uniformly random direction and has length R, ray is through a second random point chosen from within G and has length W, and ray is to a random point in ∂G and has length Y. The distribution of Y is obtained and is related to previous work. When E is specialized to G or to ∂G, other known results are retrieved. The paper ends with a discussion of a conjecture relating the means of R, W and Y.

It is shown that the convex polygons are uniquely determined (up to translation and reflection) by their covariograms. The covariogram can be represented by the ‘orientation-dependent chord length distribution', i.e. the distribution of the length of chords which are generated by random lines parallel to fixed directions. Thus the result contributes to answer Blaschke's question about the content of information comprised in chord length distributions.

While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω(n[d/2]). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d-ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d-balls is investigated for each. As n grows without bound, these expectations are O(1), O(n(d–1)/(d+4)), O(1) (for d = 2 only), and O(n(d–1)/(d+3)).

By use of a method of dissection, a formula is derived for the variance in the number of lattice points covered by a rectangle of arbitrary size, lying on a square lattice of unit spacing and inclined to the lattice at an angle of which the tangent is a rational fraction.

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.

The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(logd–1n) for any polytope, the expected number of vertices is Ω(logd–1n) for any simple polytope, and the expected number of facets is O(logd–1n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.

Let Z1, …, Zn be i.i.d. random vectors (‘points') defined in having common density f(x) that is assumed to be continuous almost everywhere. For a fixed but otherwise arbitrary norm |.| on , consider the fraction Vn of those points Z1, …, Zn that are the lth nearest neighbour (with respect to |.|) to their own kth nearest neighbour, and write Sn for the fraction of points that are the nearest neighbour of exactly k other points. We derive the stochastic limits of Vn and Sn, as n tends to∞, and show how the results may be applied to the multivariate non-parametric two-sample problem.

Every realization of a Poisson line process is a set of lines which subdivides the plane into a population of non-overlapping convex polygons. To explore the unknown statistical features of this population, an alternative stochastic construction of random polygons is developed. This construction, which is based on an alternating sequence of random angles and side lengths, provides a fast simulation method for obtaining a random sample from the polygon population. For the isotropic case, this construction is used to obtain a random sample of 2500000 polygons, providing the most precise estimates to date of some of the unknown distributional characteristics.

Recently, unbiased stereological estimators of moments of particle volume, based on measurements on lower-dimensional sections through the particles, have been developed by Jensen and Gundersen (1985). In this note, we derive an explicit form of these unbiased estimators valid for particles in Rn of ellipsoidal shape, and we establish a close relationship between the estimators and a known integral geometric formula for ellipsoids, due to Furstenberg and Tzkoni (1971). Furthermore, a stereological version of another integral geometric formula for n-dimensional ellipsoids, due to Guggenheimer (1973), is derived.

Let X1, ···, Xn be i.i.d. random variables defined in ℝd having common continuous density f(x), and let Rij be the rank of Xj in the ordered list of distances from X¡. Both the mutual neighbor probabilities p1(r, s) = P(R12 = r, R21 = s) and the neighbor-sharing probabilities p2(r, s) = P(R13 = r, R23 = s) are studied from an asymptotic viewpoint. Infinite-dimensional limits are found for both situations and take particularly simple forms. Both cases exhibit considerable stability across dimensions and thus are well approximated by their infinite-dimensional values. Tables are provided to support the results given.