The Stefan problem is a particular free boundary problem for the heat equation which arises in the investigation of the melting of solids. In the case of one space dimension there are numerous results available concerning the existence, uniqueness, and stability of the solution [c.f. 6]. However the case of several space variables is considerably more difficult. This is due in large part to the fact that the geometry of the problem can become quite complicated, and smooth initial and boundary data do not necessarily lead to smooth solutions. In particular, under heating, a connected solid can melt into two (or more) disconnected solids, thus leading to a problem in which the free boundary varies in a discontinuous manner. These difficulties have motivated several researchers to look for “weak” solutions to the Stefan problem [c.f. 1, 3, 5]. Although this approach is quite general and leads to numerical schemes for solving the problem under consideration, there are several drawbacks to this method, among them being the fact that no information is obtained concerning the structure of the interphase boundary, nor is there much information on the regularity of these weak solutions.

Let f(x) be a trigonometric polynomial with N (≥2) non-zero coefficients of absolute value not less than 1. In this paper it is proved that the L1 norm of f exceeds a fixed positive multiple of (logN/log log N)½. This result improves a previous one due to H. Davenport and P. J. Cohen (the same bound with exponent ¼).

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.

Definitions. We say that h is a Hausdorff measure function if it satisfies the following conditions:

(i) for all x > 0,

(ii) h(x) → 0 as x → 0,

(iii) h(x) is monotonic increasing.

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.