The starting point for the present paper is the following question, which asks whether points can be replaced by flats (translates of linear subspaces of arbitrary dimension) as the basic objects in a convexity structure on ℝd.

The quadratic mean of the deviation between the probability content and the interior point proportion of a random convex hull in is investigated. We obtain, in particular, an explicit and distribution-independent bound.

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK (u) is the support function of K and is the set of all unit vectors u with EhA (u) > 0. Other set-valued generalizations of the renewal function are also suggested.

One of the most beautiful and important results in geometric convexity is Hadwiger's characterization theorem for the quermassintegrals. Hadwiger's theorem classifies all continuous rigid motion invariant valuations on convex bodies as consisting of the linear span of the quermassintegrals (or, equivalently, of the intrinsic volumes) [4]. Hadwiger's characterization leads to effortless proofs of numerous results in integral geometry, including various kinematic formulas [7, 9] and the mean projection formulas for convex bodies [10]. Hadwiger's result also provides a connection between rigid motion invariant set functions and symmetric polynomials [1, 7].

In this paper we study various classes of centrally symmetric sets in d-dimensional Euclidean space Rd. As we will see, it is appropriate to focus our attention on those sets which have interior points.

In this paper, we give a sufficient condition (Theorem) in order that one domain D1 bounded by a C2-smooth boundary can be enclosed in, or enclose, another domain D0 bounded by the same kind of boundary. A same kind of sufficient condition for convex bodies (Corollary) is also obtained.

A general method for solving stereological problems for particle systems is applied to polyhedron structures. We suggested computing the kernel function of the respective stereological integral equation by means of computer simulation. Two models of random polyhedrons are investigated. First, regular prisms are considered which are described by their size and shape. The size-shape distribution of a stationary and isotropic spatial ensemble of regular prisms can be estimated from the size-shape distribution of the polygons observed in a section plane. Secondly, random polyhedrons are constructed as the convex hull of points which are uniformly distributed on surfaces of spheres. It is assumed that the size of the polyhedrons and the number of points (i.e. the number of vertices) are random variables. Then the distribution of a spatially distributed ensemble of polyhedrons is determined by its size-number distribution. The corresponding numerical density of this bivariate size-number distribution can be stereologically determined from the estimated numerical density of the bivariate size-number distribution of the intersection profiles.

Formulas for anisotropic stereology of fibre and surface processes are presented. They concern the relation between second-order quantities of the original process and its projections and sections. Various mathematical tools for handling these formulas are presented, including stochastic optimization. Finally applications in stereology are discussed, relating to intensity estimators using anisotropic sampling designs. Variances of these estimators are expressed and evaluated for processes with the Poisson property.

In this paper, two gradient properties of explicit convex functions are given.

Every homogeneous convex body in ℝd (d≥2) put to sit on a horizontal hyperplane finds a position of stable equilibrium. A cube has 2d such positions and an ellipsoid with pairwise distinct axis-lengths has 2. How many positions of stable equilibrium have most convex bodies?

The term “most” is understood in the Baire category sense. For various other results on most convex bodies, see [2], [4].

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.

A stationary (but not necessarily isotropic) Boolean model Y in the plane is considered as a model for overlapping particle systems. The primary grain (i.e. the typical particle) is assumed to be simply connected, but no convexity assumptions are made. A new method is presented to estimate the intensity y of the underlying Poisson process (i.e. the mean number of particles per unit area) from measurements on the union set Y. The method is based mainly on the concept of convexification of a non-convex set, it also produces an unbiased estimator for a (suitably defined) mean body of Y, which in turn makes it possible to estimate the mean grain of the particle process.

The typical cell of a stationary Poisson hyperplane tessellation in the d-dimensional Euclidean space is called the Poisson polytope, and the cell containing the origin is called the Poisson 0-polytope. The intention of the paper is to show that the cells of the anisotropic tessellations are in some sense larger than those of the isotropic tessellations. Under the condition of equal intensities, it is proved that the moments of order n = 1, 2, … for the volume of the Poisson 0-polytope in the anisotropic case are not smaller than the corresponding moments in the isotropic case. Similar results are derived for the Poisson polytope. Finally, generalizations are mentioned.

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a 1,· ··, an. Furthermore, let Δ(Xn ): = Vol (Bd Xn ) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn )/Ε (Δ (Xn )) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.

Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn , the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.

Let denote a rectangular lattice in the Euclidean plane E 2, 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.

It is shown that a convex body is determined uniquely among all convex bodies by the volumes of its projections onto all hyperplanes through the origin if and only if it is a parallelotope.

It is proved that for a symmetric convex body K in ℝn, if for some τ >0, |K ⋂ (x + τK) depends on ‖x‖K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies are studied.

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)).