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.

In [2], Sawyer considers a closed, central, convex region K which is such that, however it is displaced in the plane, a point of the integral lattice is covered. He shows that the area A(K) of K satisfies . We prove here a result in the opposite direction.

In this note we prove that every convex Borel set in a finite dimensional real Banach space can be obtained, starting from the compact convex sets, by the iteration of countable increasing unions and countable decreasing intersections. This question was first raised by V. Klee [1, p. 451]. It was answered affirmatively by Klee for R2 in [2, pp. 109–111] and for R3 by D. G. Larman in [4]. C. A. Rogers has given an equivalent formulation of the question for Rn in [6].

Integral geometry is the study of measures of sets of geometric figures. Commonly a measure of this sort is an integral of a density or differential form; the density is determined by the type of figure, but is independent of the particular set of such figures to which the measure is assigned. As one of the simplest examples, the area of a plane convex point set K is the integral over K of the density dx dy for points with Cartesian co-ordinates x, y. But when we assign a Hausdorff linear measure to the set of boundary points of K, we obtain a measure of quite another sort. This is representable as a Stieltjes integral of arc length density; here the density depends on the choice of K. The examples suggest examining measures for other sets of figures, where each such set is made up of all those figures from a certain class which support, in some sense, a convex body. Further, the examples lead us to expect that measures of this kind will appear as integrals of densities which may depend on the choices of . Here we treat a question of the type just described: to determine a measure for sets of q–flats which support a convex body.

In this note we shall be mainly concerned with convex bodies of constant width in En that are invariant under the group of congruences that leave invariant a regular simplex with its centre of gravity at the origin. We first show that there are many such convex bodies. This follows, by showing that any set S ot diameter 1 that is invariant under a group of congruences about the origin, is contained in a convex body of constant width 1 that is invariant under the group.

If K is a set in n-dimensional Euclidean space En, n ≥ 2, with a non-empty interior, then a point p of the interior of K is called a pseudo centre of K provided each two-dimensional flat through p intersects K in a section centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p of K is called a false centre if K is not centrally symmetric about p. Rogers [5] showed that a convex body (compact convex set with interior points) with a pseudo centre necessarily has a true centre of symmetry. But, as each interior point of an ellipsoid is a pseudo centre, the true centre need not necessarily coincide with the pseudo centre. Rogers conjectured that, for n ≥ 3, a convex body K with a false centre is necessarily an ellipsoid. In this paper we prove this conjecture.