Let d≥2 and let K⊂ℝd be a convex body containing the origin 0 in its interior. For each direction ω, let the (d−l)-volume of the intersection of K and an arbitrary hyperplane with normal ω attain its maximum when the hyperplane contains 0. Then K is symmetric about 0. The proof uses a linear integro-differential operator on Sd−1, whose null-space needs to be, and will be determined.

First, a special case of Knaster's problem is proved implying that each symmetric convex body in ℝ3 admits an inscribed cube. It is deduced from a theorem in equivariant topology, which says that there is no S4–equivariant map from SO(3) to S2, where S4 acts on SO(3) on the right as the rotation group of the cube, and on S2 on the right as the symmetry group of the regular tetrahedron. Some generalizations are also given.

For several common parent laws, the number of vertices of a sample convex hull follows a kind of law of large numbers. We exhibit an example of a parent law which contradicts a general conjecture about this matter.

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.

We study Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces, and give explicit expressions for mean surface area, mean perimeter length, and mean number of vertices of their cells. Furthermore we compare these mean characteristics with those for Poisson-Voronoi tessellations in three-dimensional Euclidean spaces. It is shown that, as the absolute value of the curvature of hyperbolic spaces increases from zero to infinity, these mean characteristics increase monotonically from those for the Euclidean case to infinity.

The intensity of a stationary process of k-dimensional affine subspaces (k-flats) of ℝd with directional distribution from a given family R is estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (the R-estimator) using the available information about the directional distribution.

Special cases are estimators for the intensity of stationary k-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in ℝd and (3) with unknown directional distribution.

We give sufficient conditions for the R-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poisson k-flat processes with directional distribution in R. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case.

Moreover, we consider stationary ergodic flat processes with directional distribution in R and general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.

Several results are proved related to a question of Steinhaus: is there a set E⊂ℝ2 such that the image of E under each rigid motion of IR2 contains exactly one lattice point? Assuming measurability, the analogous question in higher dimensions is answered in the negative, and on the known partial results in the two dimensional case are improved on. Also considered is a related problem involving finite sets of rotations.

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a packing of Euclidean 3-space with density at least 0.46421 … The best such bound previously known is 0.30051 … due to the theorem of Minkowski-Hlawka. It is probable that there is such a lower bound which is significantly greater than the one shown in this note, since there is a packing of congruent spheres which has density

The purpose of this note is to bring into attention an apparently forgotten result of C. M. Petty: a convex body has minimal surface area among its affine transformations of the same volume if, and only if, its area measure is isotropic. We obtain sharp affine inequalities which demonstrate the fact that this “surface isotropic” position is a natural framework for the study of hyperplane projections of convex bodies.

For the optimal approximation of convex bodies by inscribed or circumscribed polytopes there are precise asymptotic results with respect to different notions of distance. In this paper we derive some results on optimal approximation without restricting the polytopes to be inscribed or circumscribed.

We prove a concentration inequality for δ-concave measures over ℝn. Using this result, we study the moments of order q of a norm with respect to a δ-concave measure over ℝn. We obtain a lower bound for q∈ ]−1, 0] and an upper bound for q∈ ]0,+ ∞[ in terms of the measure of the unit ball associated to the norm. This allows us to give Kahane-Khinchine type inequalities for negative exponent.

It is shown that the cross-section body of a convex body K ⊂ ℝ3, that is the symmetric body which has for radial function in the direction u the maximal volume of a section of K by an hyperplane orthogonal to u, is a convex body in ℝ3.

Techniques currently available in the literature in dealing with problems in geometric probabilities seem to rely heavily on results from differential and integral geometry. This paper provides a radical departure in this respect. By using purely algebraic procedures and making use of some properties of Jacobians of matrix transformations and functions of matrix argument, the distributional aspects of the random p-content of a p-parallelotope in Euclidean n-space are studied. The common assumptions of independence and rotational invariance of the random points are relaxed and the exact distributions and arbitrary moments, not just integer moments, are derived in this article. General real matrix-variate families of distributions, whose special cases include the mulivariate Gaussian, a multivariate type-1 beta, a multivariate type-2 beta and spherically symmetric distributions, are considered.

An origin-symmetric convex body K in ℝn is called an intersection body if its radial function ρK is the spherical Radon transform of a non-negative measure µ on the unit sphere Sn−1. When µ is a positive continuous function, K is called the intersection body of a star body. The notion of intersection body was introduced by Lutwak [L]. It played a key role in the solution of the Busemann-Petty problem, see [G1], [G2], [L], [Z1] and [Z2]. Koldobsky [K] showed that the cross-polytope is an intersection body. This indicates that the statement in [Z3] that no origin-symmetric convex polytope in ℝn (n > 3) is an intersection body is not correct. This paper will prove the weaker statement that no origin-symmetric convex polytope in ℝn (n > 3) is the intersection body of a star body.

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in ℝd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

The paper shows that no origin-symmetric convex polyhedron in R3 is the intersection body of a star body. It is shown also that every originsymmetric convex body in Rd, for d = 3 and 4, can be seen as the intersection body of a star-shaped set whose radial function satisfies conditions related to suitable non-integer Sobolev classes.

Let X 1, X 2,… be i.i.d. random points in ℝ2 with distribution ν, and let N n denote the number of points spanning the convex hull of X 1, X 2,…,X n . We obtain lim infn→∞E (N n )n -1/3 ≥ γ1 and E (N n ) ≤ γ2 n 1/3(logn)2/3 for some positive constants γ1, γ2 and sufficiently large n under the assumption that ν is a certain self-similar measure on the unit disk. Our main tool consists in a geometric application of the renewal theorem. Exactly the same approach can be adopted to prove the analogous result in ℝd .

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.

We study dual isoperimetric deficits of star bodies. We introduce the dual Steiner ball of a star body, and use it to establish an inequality for the Lp distance, p = 2 and p = ∞, between the radial functions of two convex bodies. By applying this inequality, we find dual Bonnesen-type inequalities for convex bodies. Finally, we use a general form of Grüss's inequality to derive dual Favard-type inequalities for star and convex bodies. The results contribute to the dual Brunn–Minkowski theory initiated by E. Lutwak, and continue the attempt to understand the relation between this and the classical Brunn–Minkowski theory.

If K is a convex body in d and 1≤k≤d − 1, we define Pk(K) to be the Minkowski sum or Minkowski average of all the projections of K onto k-dimensional subspaces of d. The operator Pd − 1, was first introduced by Schneider, who showed that, if Pd − 1(K) = cK, then K is a ball. More recently, Spriestersbach showed that, if Pd − 1(K) = cK then K = M. In addition, she gave stability versions of this result and Schneider's. We will describe further injectivity results for the operators Pk. In particular, we will show that Pk is injective if k≤d/2 and that P2 is injective in all dimensions except d = 14, where it is not injective.