For some integer modulus q let F be a subset of Z(q), the additive group of residues modulo q. As in «1» we shall write

and

Let K3 be a non-Galois cubic extension of the rationals and let K6 be its normal closure. Under K6 there is a unique quadratic field K2. For i = 2, 3, 6 we define C1i; to be the 3-class group of K2 and ri to be the rank of Cli. We suppose that K2 is complex and that K6/K2 is unramified. Our main result is

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.

§1. Introduction and Summary. Throughout X is a complete separable metric space. We write K1 for the family of non-empty compact subsets of X. K1 may be endowed with a metric (first introduced by Hausdorff) under which K1 is complete and separable. We shall make use of the subbase for this metrizable topology of K1 given by sets of the two forms

for U open in X (see Kuratowski [4] or E. Michael [9] for a discussion of topologies on the space of subsets of X). if we shall be concerned with sets in [0, 1] × X which are universal for ℋ. To define these let us make the convention that, for D ⊆ [0, 1] × X, we write

D is said to be universal for ℋ if

Using toroidal co-ordinates, an exact solution is derived for the velocity field induced in two immiscible semi-infinite viscous fluids, possessing a plane interface, by the slow rotation of a concave spherical lens, which is such that the circle of intersection of the composite spherical surfaces lies in the plane of the interface. The expression for the torque acting on the lens is derived, and this is shown to be reducible to an analytic closed form, when the lens degenerates into a spherical bowl and the fluids are identical.

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].

In this paper I prove the following result:

Theorem. Let f(d) be a multiplicative function such that |f(d)| ≤ 1 and ∑{ l/p : p ε P} = ∞, where P denotes the set of primes p for which f{p) = −1, and let v(n) denote the number of distinct prime factors ofn. Then for almost all n,

where A is an arbitrary constant > 3/e.

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.