We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S2 problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S2 problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.

The covariogram gK(x) of a convex body K ⊆ Ed is the function which associates to each x ∈ Ed the volume of the intersection of K with K + x, where Ed denotes the Euclidean d-dimensional space. Matheron (1986) asked whether gK determines K, up to translations and reflections in a point. Positive answers to Matheron's question have been obtained for large classes of planar convex bodies, while for d ≥ 3 there are both positive and negative results. One of the purposes of this paper is to sharpen some of the known results on Matheron's conjecture indicating how much of the covariogram information is needed to get the uniqueness of determination. We indicate some subsets of the support of the covariogram, with arbitrarily small Lebesgue measure, such that the covariogram, restricted to those subsets, identifies certain geometric properties of the body. These results are more precise in the planar case, but some of them, both positive and negative ones, are proved for bodies of any dimension. Moreover some results regard most convex bodies, in the Baire category sense. Another purpose is to extend the class of convex bodies for which Matheron's conjecture is confirmed by including all planar convex bodies possessing two nondegenerate boundary arcs being reflections of each other.

Recently, systematic sampling on the circle and the sphere has been studied by Gual-Arnau and Cruz-Orive (2000) from a design-based point of view. In this note, it is shown that their mathematical model for the covariogram is, in a model-based statistical setting, a special case of the p-order shape model suggested by Hobolth, Pedersen and Jensen (2000) and Hobolth, Kent and Dryden (2002) for planar objects without landmarks. Benefits of this observation include an alternative variance estimator, applicable in the original problem of systematic sampling. In a wider perspective, the paper contributes to the discussion concerning design-based versus model-based stereology.

Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.

Knowing the (geometric) covariogram of a convex body is equivalent to knowing, for each direction u, the distribution of the lengths of the chords of that body which are parallel to u. We prove that the covariogram determines convex polygons, among all convex bodies, up to translation and reflection. This gives a partial answer to a problem posed by Matheron.

Useful approximations have been developed along the years to predict the precision of systematic sampling for measurable functions of a bounded support in ℝd. Recently, the theory of systematic sampling on ℝ has received a thrust. In geometric sampling, design based unbiased estimators exist, however, which imply systematic sampling on the circle (𝕊1) and the semicircle (ℍ1); the planimeter estimator of an area, or the Buffon-Steinhaus estimator of curve length in the plane constitute popular examples. Over the last two decades, many other estimators of geometric measures have been obtained which imply systematic sampling also on the sphere (𝕊2). In this paper we adapt the theory available for non-periodic functions of bounded support on ℝ to periodic functions, and thereby to 𝕊1 and ℍ1, and we obtain new estimators of the corresponding variance approximations. Further we consider - we believe for the first time - the problem of predicting the precision of systematic sampling in 𝕊2. The paper starts with a historical perspective, and ends with suggestions for further research.

In design stereology, and in the context of geometric sampling in general, the problem often arises of estimating the integral of a bounded non-random function over a bounded manifold D ⊂ ℝn by systematic sampling with geometric probes. Variance predictors, often based on Matheron's theory of regionalized variables, are available when the relevant function is sampled at the points of a grid intersecting D, but not when the dimension of the probes is greater than zero. For instance, the volume of a bounded object may be estimated using parallel systematic planes, which amounts to sampling on ℝ1 with systematic points, or using parallel systematic slabs of thickness t > 0, which amounts to sampling on ℝ1 with non-overlapping systematic segments of length t > 0. Useful variance predictors exist for the former case, but not for the latter. In this paper we set out a general scheme to predict estimation variances when the dimension of either D, or of the probes, is n. We make some progress when both dimensions are equal to n, and obtain explicit results for n = 1 (e.g. for systematic slice sampling). We check and illustrate our results for the volume estimators of ellipsoids and of rat lung.

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.