We give here some extensions of inequalities of Popoviciu and Rado. The idea is to use an inequality [C. P. Niculescu and L. E. Persson, Convex functions. Basic theory and applications (Universitaria Press, Craiova, 2003), Page 4] which gives an approximation of the arithmetic mean of n values of a given convex function in terms of the value at the arithmetic mean of the arguments. We also give more general forms of this inequality by replacing the arithmetic mean with others. Finally we use these inequalities to establish similar inequalities of Popoviciu and Rado type.

We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝd by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.

Generalized local mean normal measures μz , z ∈ R d , are introduced for a nonstationary process X of convex particles. For processes with strictly convex particles it is then shown that X is weakly stationary and weakly isotropic if and only if μz is rotation invariant for all z ∈ R d . The paper is concluded by extending this result to processes of cylinders, generalizing Theorem 1 of Schneider (2003).

Let be a family of axis-aligned parallelotopes, or boxes, in ℝd. Denote by fk () the number of subfamilies of of size k + 1 with non-empty intersection. In an earlier paper, the author proved that, if f0 () = n and fr() = 0, then fk() ≤ fk(n, d, r) for k = 1,…,r − 1, where fk(n, d, r) is some explicitly given number. The result is best possible for all k. Here it is shown that, if equality is attained for some such k, then equality is attained for each such k.

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝn which is closed under Minkowski addition and non-negative dilatations. A convex body in ℝn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1, T2 such that M + T1 = T2, and T1, T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

Variations and generalizations of several classical theorems concerning characterizations of ellipsoids are developed. In particular, these lead to a short and comprehensible proof of the false centre theorem.

Let K be a convex body of dimension at least 3, and let p0 be a point. If every section of K through p0 is centrally symmetric, then Rogers proved in [6] that K is centrally symmetric, although p0 may not be the centre of K. If this is the case, then Aitchison, Petty and Rogers [1] and Larman [2] proved that K must be an ellipsoid. Suppose now that, for every direction, we can choose continuously a section of K that is centrally symmetric; if K is strictly convex, then Montejano [3] proved that K must be centrally symmetric. Consider now the following example. Let D be a (euclidean) ball centred at the origin from which two symmetric caps are deleted. Then D is centrally symmetric with respect to the origin, and has a lot of circular sections whose centre is not the origin. In fact, we can choose continuously, for every direction, a section of D which is centrally symmetric, in such a way that not all these sections pass through the origin. Nevertheless, no matter how we choose these sections, there are always necessarily many of them that do pass through the origin. For those sections, of course, we have not imposed any condition, which explains the fact that D is not a quadric elsewhere.

The kth projection function of a convex body K ⊂ ℝn assigns to any k-dimensional linear subspace of ℝn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in ℝn, and let K0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K0 with ∂K0 of class C2 with positive radii of curvature). Assume that K and K0 have proportional 1st projection functions (i.e., width functions) and proportional kth projection functions. For 2 ≤ k < (n + 1)/2 and for k = 3, n = 5, it is shown that K and K0 are homothetic. In the special case where K0 is a Euclidean ball, characterizations of Euclidean balls as convex bodies of constant width and constant k-brightness are thus obtained.

The covariogram g K (x) of a convex body K ⊆ E d is the function which associates to each x ∈ E d the volume of the intersection of K with K + x, where E d denotes the Euclidean d-dimensional space. Matheron (1986) asked whether g K 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.

In this paper, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on the metric addition inequality of Alexander.

2000 Mathematics subject classification: primary 52A40.

Intersection densities are introduced for a large class of nonstationary Poisson processes of hypersurfaces and inequalities for them are proved. In doing so, similar results from both Wieacker (1986) and Schneider (2003) are summarized in one theorem and the concept of an associated zonoid of a Poisson process of hypersurfaces is generalized to a nonstationary setting.

The volume fraction of the intact grains of the dead leaves model with spherical grains of equal size is 2−d in d dimensions. This is the volume fraction of the original Stienen model. Here we consider some variants of these models: the dead leaves model with grains of a fixed convex shape and possibly random sizes and random orientations, and a generalisation of the Stienen model with convex grains growing at random speeds. The main result of this paper is that if the radius distribution in the dead leaves model equals the speed distribution in the Stienen model, then the volume fractions of the two models are the same in this case also. Furthermore, we show that for grains of a fixed shape and orientation, centrally symmetric sets give the highest volume fraction, while simplices give the lowest. If the grains are randomly rotated, then the volume fraction achieves its highest value only for spheres.

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

Assume that n points P1,…,Pn are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P1,…,Pn, and (1, 0). The vertices of the convex hull form a convex chain. Let be the probability that the convex chain consists – apart from the points (0, 1) and (1, 0) – of exactly k of the points P1,…,Pn. Bárány, Rote, Steiger, and Zhang [3] proved that . The values of are determined for k = 1,…,n − 1, and thus the distribution of the number of vertices of a random convex chain is obtained. Knowing this distribution provides the key to the answer of some long-standing questions in geometrical probability.

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in ℝ n with smaller i-dimensional central sections necessarily have smaller volume. A generalization of this problem is studied, when the volumes are measured with weights satisfying certain conditions. The case of hyperplane sections (i = n − 1) has been studied by A. Zvavitch.

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.

We study a random field obtained by counting the number of balls containing a given point when overlapping balls are thrown at random according to a Poisson random measure. We describe a microscopic process which exhibits multifractional behavior. We are particularly interested in the local asymptotic self-similarity (LASS) properties of the field, as well as in its X-ray transform. We obtain two different LASS properties when considering the asymptotics either in law or in the sense of second-order moments, and prove a relationship between the LASS behavior of the field and the LASS behavior of its X-ray transform. These results can be used to model and analyze porous media, images, or connection networks.

We compare the geometric concept of strict convexity of open subsets of Rn with the analytic concept of 2-strict convexity, which is based on the defining functions of the set, and we do this by introducing the class of 2N-strictly convex sets. We also describe an exhaustion process of convex sets by a sequence of 2-strictly convex sets.

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp-extension of such a definition makes use of the sum of the pth power of the support functions. An Lp-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of a subspace of ℓq, where 1/p + 1/q = 1. In this paper, a sharp upper estimate is given of the volume of Zp in terms of the volume of Z1, as well as a sharp lower estimate of the volume of the polar of Zp in terms of the same quantity. In particular, for p = 1, the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.

A tensor-type integral formula for intrinsic volumes is used to define a further variant of directed projection functions and show that these determine a convex body uniquely. Averages of directed projection functions are then studied, and the connections between the resulting operators and previously considered spherical transforms discussed.