If C is a convex open set of a Hausdorff locally convex space, then for every compact set K ⊂ C, the closed convex hull of K is contained in C. It follows that the same is true if C is a countable intersection of convex open sets of E. However, as it is well known (and also follows from §7(a)) a convex of Gδ is not always a countable intersection of open convex sets. This paper is devoted to showing that the Gδ convex sets have some remarkable properties.

A Borel isomorphism that, together with its inverse, maps ℱσ-sets to ℱσ-sets will be said to be a Borel isomorphism at the first level. Such a Borel isomorphism will be called a first level isomorphism, for short. We study such first level isomorphisms between Polish spaces and between their Borel and analytic subsets.

We say a motion g brings a mobile convex body K into inner contact with a fixed body K0 if the image gK lies in K0 and shares a boundary point with K0; we speak of the inner contact being at the common boundary point. The mobile body K is said to roll freely in K0 if, corresponding to each boundary point x of K0 and each rotation R, there is a translation t such that RK + t = gK has inner contact with K0 at x.

We are concerned with invertible transformations of the unit n-dimensional cube In, 2 ≤ n ≤ ∞, which preserve n-dimensional Lebesgue measure μ. Following Halmos [4], we denote the space of all such transformations by G = G(In), and the subset of G consisting of homeomorphisms by M = M(In). We ask to what extent, and in what sense, can we approximate an arbitrary transformation g in G by a homeomorphism h in M. New results are obtained in the course of presenting a new proof of the theorem of J. Oxtoby and H. E. White, Jr., stated below.

This paper originated with the observation that while all of the known stable lattice packings of spheres are highly symmetric, it is futile to try to prove a converse statement: the ordinary integer-lattice provides a distinctly unstable packing of spheres, but admits a large group of orthogonal symmetries nonetheless. The integerlattice is in fact very unstable—the slightest perturbation places the spheres in a more efficient configuration. We will call such a lattice fragile. The purpose of this note is to prove that a highly symmetric lattice must be either stable or fragile.

In an article generalising work of Roquette and Zassenhaus, Connell and Sussman [2] have demonstrated the importance of certain prime ideals in a number field k0 for estimating the l-rank of the class group of an extension k. These ideals have a power prime to l which is principal and all their prime factors in k have ramification index divisible by l. The products of the prime divisors of these ideals in the normal closure K of k/k0 are invariant under Gal (k/k0). Thus certain roots in k of the ideals in k0 are in some sense fixed by the Galois group. This leads to the concept of ambiguous ideals in an extension k/k0 which is not necessarily normal.

We show that a complete metric space X has an essentially unique “nice” zero-dimensional dense Gδ subset, and derive from this a complete algebraic description of the “category algebra” (of Borel modulo first category sets) of X.

G. Higman [5] first considered conditions on a group G sufficient to ensure that for any ring R with no zero-divisors the group-ring RG contains no zero-divisors. It has been shown by various authors that if G belongs to one of the classes of locally indicible groups [5], right-ordered groups [6], polycyclic groups [4] or positive one-relator groups [1] then it is enough that G should be torsionfree. The proofs rely heavily on the special properties of the classes of groups involved but it may be conjectured that it is a sufficient condition in general that G should be torsionfree and no counterexamples are known.

Let K be any finite (possibly trivial) extension of ℚ, the field of rational numbers. Let denote the ring of integers of K, and let M ⊆ be a full module in K thus a free ℤ-module of rank [K : ℚ] contained in ; ℤ denoting the ring of rational integers. Regarding as an abelian group, the index (: M) is finite. Suppose that m1, …, mk is a ℤ-basis for M and let a ∊ Then the polynomial

(the xi; being indeterminates) will be called a full-norm polynomial; here NK/ℚ denotes the norm mapping from K to ℚ. Apart from constant factors, such a polynomial f(x) is necessarily irreducible in ℤ[x].

A generalization and simplification of F. John's theorem on ellipsoids of minimum volume is proven. An application shows that for 1 ≤ p < 2, there is a subspace E of Lp and a λ > 1 such that 1E has no λ-unconditional decomposition in terms of rank one operators.

Lyndon's axiomatic methods are used in [1] to show, among other things, that a group G with an integer valued length function satisfying certain conditions is free. At the end of his paper [2] Lyndon gives a method of embedding such a group in a free group whose natural length function extends the function on G. We construct here a simpler embedding with the same property.

Suppose that we have a system of congruences ai (mod ni) 1 < n1 < … < ni < … < nk such that every integer is congruent to at least one ai (mod ni), then we say that it is a covering system of congruences. If ni | m, 1 ≤ i ≤ k, we say that m is a covering number. We shall use the symbol ℕ to denote the natural numbers together with zero, then m is a covering number if, for each q there is an aq such that

The Picard group P(ZG) of the integral group ring ZG is defined as the class group of two-sided invertible ZG-ideals of QG modulo those principal ideals generated by an invertible central element. The basic properties of Picard groups have been established by A. Fröhlich, I. Reiner and S. Ullom [1], [2], [3]. In this note we settle an outstanding question by exhibiting a class of finite p-groups G whose Picard groups contain nontrivial elements which are represented by principal ideals; these elements remain nontrivial in P(ZpG) also. We obtain these ideals from outer automorphisms of the groups.