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)).

This paper considers sets of points from a Poisson process in the plane, chosen to be close together, and their properties. In particular, the perimeter of the convex hull of such a point set is investigated. A number of different models for the selection of such points are considered, including a simple nearest-neighbour model. Extensions to marked processes and applications to modelling animal territories are discussed.

Let K be a convex body in Euclidean space Rd, d≥2, with volume V(K) = 1, and n ≥ d +1 be a natural number. We select n independent random points y1, y2, …, yn from K (we assume they all have the uniform distribution in K). Their convex hull co {y1, y2, …, yn} is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope

It is easy to see that if U: Rd → Rd is a volume preserving affine transformation, then for every convex body K with V(K) = 1, m(K, n) = m(U(K), n).

Let A and B be two compact, convex sets in ℝn, each symmetric with respect to the origin 0. L is any (n - l)-dimensional subspace. In 1956 H. Busemann and C. M. Petty (see [6]) raised the question: Does vol (A ⌒ L) < vol (B ⌒ L) for every L imply vol (A) < vol(B)? The answer in case n = 2 is affirmative in a trivial way. Also in 1953 H. Busemann (see [4]) proved that if A is any ellipsoid the answer is affirmative. In fact, as he observed in [5], the answer is still affirmative if A is an ellipsoid with 0 as center of symmetry and B is any compact set containing 0.

The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Here we show corresponding inequalities for arbitrary convex bodies, where the successive minima are replaced by certain successive diameters and successive widths.

We further give some applications of these results to successive radii, intrinsic volumes and the lattice point enumerator of a convex body.

In [7] the notion of minimal pairs of convex compact subsets of a Hausdorff topological vector space was introduced and it was conjectured, that minimal pairs in an equivalence class of the Hörmander-Rådström lattice are unique up to translation. We prove this statement for the two-dimensional case. To achieve this we prove a necessary and sufficient condition in terms of mixed volumes that a translate of a convex body in ℝn is contained in another convex body. This generalizes a theorem of Weil (cf. [10]).

This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.

Let K ⊂ Rd be a convex body and choose points xl, x2, …, xn randomly, independently, and uniformly from K. Then Kn = conv {x1, …, xn} is a random polytope that approximates K (as n → ∞) with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K – vol Kn when K is a smooth convex body. Moreover, this result is extended to quermassintegrals (instead of volume).

A continuous selection and a coincidence theorem are proved in H-spaces which generalize the corresponding results of Ben-El-Mechaiekh-Deguire-Granas, Browder, Ko-Tan, Lassonde, Park, Simon and Takahashi to noncompact and/or nonconvex settings. By applying the two theorems, some intersection theorems concerning sets with H-convex sections are obtained which generalize the corresponding results of Fan, Lassonde and Shih-Tan to H-spaces. Some applications to minimax principle are given.

On a convex surface S ⊂ Rd, two points x, y are conjugate if there are at least two shortest paths, called segments, from x to y. This paper is about the set of points conjugate to some fixed point xєS.

The main result of this paper is the following theorem. If P is a convex polytope of Ed with affine symmetry, then P can be illuminated by eight (d - 3)-dimensional affine subspaces (two (d- 2)-dimensional affine subspaces, resp.) lying outside P, where d ≥ 3. For d = 3 this proves Hadwiger's conjecture for symmetric convex polyhedra namely, it shows that any convex polyhedron with affine symmetry can be covered by eight smaller homothetic polyhedra. The cornerstone of the proof is a general separation method.

The purpose of this work is to investigate the relationship between Radon transforms and centrally symmetric convex bodies. Because of the injectivity properties of the Radon transform it is natural to consider transforms on the sphere separately from those on the higher order Grassmannians. Here we shall concentrate on the latter, whilst the former will be the subject of another article presently in preparation, Goodey and Weil [1991].

Let K1, K2,…, Kd+1 be d+1 convex bodies in Ed, the interiors of which have no point in common. If mX denotes the measure of set X, we prove

and this inequality is best possible.

Let

It is proved that for suitable a and b, n≥7, one can have Vn(An) = Vn(Bn) and for every (n–1)-dimensional subspace H of ℝn, where Bn is the unit ball of ℝn. This strengthens previous negative results on a problem of H. Busemann and C. M. Petty.

A convex compact subset of ℝd is called a convex body. The (Euclidean) surface area and volume of a convex body K are denoted s(K) and v(K) respectively. The support function of a convex body K is denned by h(K, x) = maxy∈K xty and the polar dual of K is given by K0 = {x: |xty|1, y∈K}. Double vertical bars shall denote the Euclidean length of a vector , and S shall denote the unit sphere (the Euclidean unit ball): S = {x: ║x║≤1}. We use for the mixed volume

The aim of this paper is to prove the following

THEOREM. Let M be a C∞ compact and strictly convex surface embedded in the euclidean space E3 or in the hyperbolic space H3. We suppose that all shadow-lines ofM are congruent. Then M is a euclidean 2-sphere or a hyperbolic 2-sphere respectively.

Let M be a convex body, i.e., a compact, convex set with non-empty interior, in n-dimensional Euclidean space En. A chord [a, b] of M is said to be an affine diameter of M, if, and only if, there exists a pair of (different) parallel supporting hyperplanes of that body, each containing one of the points a, b. The following result of Eggleston (cf. [1] and [2]) is well-known. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to exactly three affine diameters. In [3] this result is sharpened. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to at least two, but a finite number of affine diameters. A natural problem for the n-dimensional case, based on Eggleston's result, is the following (cf. also [4]). Is it true that the n-dimensional simplex is the only convex body in En such that through each interior point pass precisely 2n − l affine diameters? For the case of convex polytopes, i.e., convex bodies with a finite number of extreme points, we shall give a positive answer to this question.