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.

A sequence {an} of integers is said to be primitive if ai × aj whenever i ≠ j. Let f be a polynomial with integer coefficients and A a sequence of positive integers. We discuss further a problem considered in [1] in which I. Anderson, W. W. Stothers and the author investigated primitive sequences of the form f(A) = {f(x), x ∈ A}. (Of course, we can assume f(x)→ ∞ as x → ∞.) We shall prove the following theorem in which A(n), as usual, denotes the number of memhers of A that are. less than or equal to n.

A centrally symmetric d-polytope (d-dimensional convex polytope) P in Euclidean space Ed is called k-neighbourly provided every subset of k vertices of P, which does not contain two opposite vertices of P, is the set of vertices of a (k − 1)-simplex which is a face of P. Contrasting the situation of neighbourly polytopes without the symmetry assumption (see, e.g., Griinbaum [1; chap. 7]), it appears that the possible neighbourliness properties of centrally symmetric polytopes are rather restricted. For d ≥ 2 and n ≥ 1, let k(d, n) denote the greatest integer k, such that there exists a k-neighbourly, centrally symmetric d-polytope with 2(d + n) vertices. McMullen and Shephard [4] have shown that , and for n ≥ 3. They conjectured that

Hitherto, little use has been made of the well-known representation

of the Riemann zeta-function, ζ(s), within the critical strip l = {s:s = σ + it, 0 < σ < 1}. Certain variants of (1) have been used to deduce the functional equation of ζ(s), while a simple consequence of (1) itself is that ζ(s) does not vanish on the positive real axis.

In this paper we continue our investigation of the topological filtration on the complex representation ring R(G) of a finite group, see [4] and [5]. To recall the basic definitions from (1): let

map a k-dimensional representation ζ to the (flat) vector bundle over the classifying space BG associated to the universal G-bundle by the G-structure on Ck. Then, if denotes the (2k − l)-skeleton of BG,

Let x, y, Q denote large real numbers, with x > y > Q. The object of this paper is to prove the following result

for arbitrary A > 0, with a larger value of Q than hitherto. The symbol denotes as usual the suppression of an absolute constant, π(N; q, a) denotes the number of prime numbers up to N which are congruent to a (mod q), and ϕ(q) denotes Euler's function.

Let G be any enumerable subset of the positive real numbers, with infinity as its only limit point. The purpose of this paper is to give a construction for a Lebesgue measurable set E ⊂ R+, with the following properties:

A d-dimensional zonotope Z in Ed which is the vector sum of n line segments is linearly equivalent to the image of a regular n-cube under some orthogonal projection. The zonotope in En-d which is the image of the same cube under projection on to the orthogonal complementary subspace is said to be associated with Z. In this paper is proved a conjecture of G. C. Shephard, which asserts that, if Z tiles Ed by translation, with adjacent zonotopes meeting facet against facet, then tiles En-d in the same manner. A number of conditions, conjectured by Shephard and H. S. M. Coxeter to be equivalent to the tiling property, are also proved.