Let γ:x = x(u) for a≤ u ≤ b be a closed curve in n dimensional euclidean space En (for n ≥ 2) referred to some point P0, which does not lie on γ, as origin. We suppose that γ is smooth in the sense that the cartesian coordinates are class C2 functions of the parameter u, and that dx/du is non-vanishing so that a tangent vector is everywhere well defined. These properties are also assumed to hold in an obvious way at the join of the end points u = a and u = b. As P moves on γ its position vector x(u) intersects the surface of the unit sphere centred at P0 in a closed curve γ0. Note that γ0: x = x0(u) may not be smooth everywhere. If there are points P on γ where P0P is tangential to γ at P then dx0/du =0 at the corresponding point on γ0, and γ0 may have a cusp there. We assume that γ0 is smooth except at a finite number of points. We define the total swing of the position vector to γ to be the arc length L0 of γ0. Clearly L0 is not an invariant but depends on the choice of the origin P0 in relation to γ. The total (first) curvature of γ is an invariant and is defined by

where s is the arc length on γ, K(S) is the curvature and ‖v‖ is the euclidean norm of the vector v. Note that LT is also the arc length of the closed curve γ1, described on a unit sphere by the unit tangent t = dx/ds to γ as position vector, with the centre of the sphere as origin, γ1 is the spherical indicatrix of t. Our purpose is to establish the result

for all choices of P0.

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.

We shall say that the sets A, B ⊂ Rk are equivalent, if they are equidecomposable using translations; that is, if there are finite decompositions and vectors x1,…, xd∈Rk such that Bj = Aj + xj, (j = 1,…,d). We shall denote this fact by In [3], Theorem 3 we proved that if A ⊂ Rk is a bounded measurable set of positive measure then A is equivalent to a cube provided that Δ(δA)<k where δA denotes the boundary of A and Δ(E) denotes the packing dimension (or box dimension or upper entropy index) of the bounded set E. This implies, in particular, that any bounded convex set of positive measure is equivalent to a cube. C. A. Rogers asked whether or not the set

An oversight by the typesetters has led to the omission of the summation condition on the second summation of the first displayed equation on page 392 of [1]. The proof of Lemma 4.1 is consequently very difficult to follow. The expression for Ip(α) should read

In addition, we take this opportunity to note that on page 391, in the two displayed equations following equation (3.15), the summations involving ud+1 should read

Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given.

Let ℳ denote the class of all finite positive Borel measures μ in Rn with compact support. We define the Fourier transform of μεℳ by writing

and the α-energy of μ is given by

General expressions are found for the orthonormal polynomials and the kernels relative to measures on the real line of the form μ + Mδc, in terms of those of the measures dμ and (x − c)2dμ. In particular, these relations allow us to show that Nevai's class M(0, 1) is closed under adding a mass point, as well as obtain several bounds for the polynomials and kernels relative to a generalized Jacobi weight with a finite number of mass points.

The analytic paracommutators in the periodic case have been studied. Their boundedness, compactness, the Schatten-von Neumann properties and the cut-off phenomena have been proved. These results have been applied to some kind of operators on the Bergman spaces that have cut-off at any p∈(0, ∞).

In this paper we shall be concerned with the growth of functions in the class ℳp, where we write f ∈ ℳp 1 <p<∞, if:

(i) f is absolutely continuous on bounded subintervals of [0, ∞]);

(ii) f(0) = 0; and

(iii) .

In this paper the Hausdorff dimension of systems of real linear forms which are simultaneously small for infinitely many integer vectors is determined. A system of real linear forms,

where ai, xij∈ℝ, 1 ≤i≤m, 1≤j≤n will be denoted more concisely as

where a∈⇝m, X∈ℝmn and ℝmn is identified with Mm × n(ℝ), the set of real m × n matrices. The supremum norm of any vector in k dimensional Euclidean space, ℝk will be denoted by |v|. The distance of a point a from a set B, will be denoted by dist (a, B) = inf {|a − b|: b ∈ B}.