From time to time results about partitioning a given set into subsets have been established. (See for example [2], [3].) We consider here the reverse problem of forming the union of three sets in a certain best possible way. For simplicity we work in Euclidean n-space, En. Let mX denote the measure of the set X.

A convex polytope is a zonotope, if, and only if, its support function satisfies Hlawka's inequality. It follows that a finite dimensional real space with piecewise linear norm is isometrically isomorphic to a subspace of an L1 space, if, and only if, it has the quadrilateral property

Consider two convex bodies K, K′ in Euclidean space En and paint subsets β, β′ on the boundaries of K and K′. Now assume that K′ undergoes random motion in such a way that it touches K.

Let denote the class of all compact convex sets in Euclidean n-dimensional space En, and let y be the collection of those members of k which are centrally symmetric. The topology in is that induced by the Hausdorff metric.

A finite family (Ci)i ∊ I of at least two convex subsets of ℝn is said to have the intersection property provided that the set is non-empty for all families (ai)i ∊ I of points in ℝn. Previously, D. G. Larman [2, Theorem 3] has given a sufficient condition (which is not necessary) and an “almost” necessary condition (which is not sufficient) for to have the intersection property.

A quasi-simplex of ℝd is the closure of a Choquet simplex of ℝd. We characterize these quasi-simplices and we use this characterization to describe the line-free Choquet simplices of ℝd.

Let K, K′ be two centrally symmetric convex bodies in En, with their centres at the origin o. Let Vr denote the r-dimensional volume function. A problem of H. Busemann and C. M. Petty [1], see also, H. Busemann [2] asks:—

“If, for each (n − 1)-dimensional subspace L of En,

does it follow that

If n = 2 or, if K is an ellipsoid, then Busemann [3] shows that it does follow. However we will show that, at least for n ≥ 12, the result does not hold for general centrally symmetric convex bodies K, even if K′ is an ellipsoid. We do not construct the counter example explicitly; instead we use a probabilistic argument to establish its existence.

If K is a set in n-dimensional Euclidean space En, n ≥ 2, with non-empty interior, then a point p of En is called a pseudo-centre of K provided tha each two dimensional flat through p intersects K in a section, which is either empt or centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p is called a false centre of K, if K is not centrally symmetric about p. A dee result of Aitchison, Petty and Rogers [1] asserts that, if K is a convex body in E and p is a false centre of K, with p in the interior, int K, of K, then K is an ellipsoid Recently J. Höbinger [2] extended this theorem to any smooth convex body K witl a false centre p anywhere in En. At a recent meeting in Oberwolfach, he asked if the condition of smoothness can be omitted and the purpose of this note is to prove sucl a result.

Let Zn denote the set of all ordered n-tuples of integers. Let us call any finite subset of Zn a body in Zn, and any finite set of bodies in Zn a family in Zn.

Consider the following problem:

Give a decision procedure which for any family ℱ in Zn decides the following.

In [3] it was proved that every convex Borel (= Baire) set in a finite dimensional real Banach space can be obtained, starting from the closed (or compact) convex sets, by the iteration of countable increasing unions and countable decreasing intersections.

In §2 of this note we define some concepts of the descriptive theory of convex sets in locally convex spaces. We prove several theorems, which are analogous to the standard theorems of the descriptive theory of sets in topological spaces.

1. W. Blaschke's kinematic formula in the integral geometry of Euclidean n-dimensional space gives a weighted measure to the set of positions in which a mobile figure K1 overlaps a fixed figure K0. In the simplest case, K0 and K1 are compact convex sets and all positions are equally weighted; we give this in more detail. Let Wq denote the q-th Quermassintegral of K1: Steiner's formula for the volume V of the vector sum K1 + λB of K1 and a ball of radius λ defines these set functions by the equation

see [4; p. 214]. Blaschke's formula [4; p. 243] gives

as the measure, to within a normalization, of overlapping positions of K1 relative to K0.

Often stones on beaches pounded by waves wear into quite smooth, regular shapes, sometimes apparently ellipsoidal and even spherical [8]. This paper begins with an idealization of this wearing process for materials isotropic with respect to wear, then develops an equation governing the idealized process, and goes on to show that a stone which is initially convex and centrally symmetric tends to assume a spherical shape as a consequence of the governing equation. This conclusion is predicated on the assumption that the mathematical conditions describing the wearing process are those of a well-posed problem.

1. A convex body K in n-dimensional Euclidean space En (nonvoid, compact, and convex subset of En) is uniquely determined, up to translations, by its mixed volumes with other convex bodies. Therefore one might expect that relations between two convex bodies K and L correspond to relations between the mixed volumes of K, L, and other bodies. In this paper we give conditions, formulated in terms of mixed volumes, which are necessary and sufficient for a certain property of decomposability, namely, the property that L is a summand of K, and we solve some related problems.

Introduction. All the sets X considered in this paper are assumed to be compact convex sets in the Euclidean plane. We shall let K denote this class of sets. Problems concerning the division of such sets by three non-concurrent lines have been considered by Eggleston [1] on page 118 and by Grünbaum [2].

Asymmetry classes of convex bodies have been introduced and investigated by G. Ewald and G. C. Shephard [2], [3], [6]. These classes are defined as follows. Let denote the set of all convex bodies in n-dimensional Euclidean space ℝn. For K1, K2 ∊ write K1 ∼ K2 if there exist centrally symmetric convex bodies S1, S2 ∊ such that

where + denotes Minkowski addition. Then ∼ is an equivalence relation on and the corresponding classes are called asymmetry classes. The asymmetry class which contains K is denoted by [K].

Blaschke [1] introduced the notion of maximal tetrahedra inscribed in two and three dimensional convex sets (maximal in the sense of volume). From this notion, he derived an inequality relating the volume of such maximal tetrahedra and the volume of the convex set, and used the inequality to characterize an ellipsoid and to obtain some results concerning isoperimetric inequalities.

In [2], Sawyer considers a closed, central, convex region K which is such that, however it is displaced in the plane, a point of the integral lattice is covered. He shows that the area A(K) of K satisfies . We prove here a result in the opposite direction.

In this note we prove that every convex Borel set in a finite dimensional real Banach space can be obtained, starting from the compact convex sets, by the iteration of countable increasing unions and countable decreasing intersections. This question was first raised by V. Klee [1, p. 451]. It was answered affirmatively by Klee for R2 in [2, pp. 109–111] and for R3 by D. G. Larman in [4]. C. A. Rogers has given an equivalent formulation of the question for Rn in [6].

Integral geometry is the study of measures of sets of geometric figures. Commonly a measure of this sort is an integral of a density or differential form; the density is determined by the type of figure, but is independent of the particular set of such figures to which the measure is assigned. As one of the simplest examples, the area of a plane convex point set K is the integral over K of the density dx dy for points with Cartesian co-ordinates x, y. But when we assign a Hausdorff linear measure to the set of boundary points of K, we obtain a measure of quite another sort. This is representable as a Stieltjes integral of arc length density; here the density depends on the choice of K. The examples suggest examining measures for other sets of figures, where each such set is made up of all those figures from a certain class which support, in some sense, a convex body. Further, the examples lead us to expect that measures of this kind will appear as integrals of densities which may depend on the choices of . Here we treat a question of the type just described: to determine a measure for sets of q–flats which support a convex body.

In this note we shall be mainly concerned with convex bodies of constant width in En that are invariant under the group of congruences that leave invariant a regular simplex with its centre of gravity at the origin. We first show that there are many such convex bodies. This follows, by showing that any set S ot diameter 1 that is invariant under a group of congruences about the origin, is contained in a convex body of constant width 1 that is invariant under the group.