C. A. Rogers

The realization of distances within sets in Euclidean space

In 1944 and 1945 H. Hadwiger [1, 2] proved the following theorems. Theorem A. Let E n be covered by n + 1 closed sets. Then there is one of the sets, within which all distances are realized. Theorem B. Let E n be covered by 4n−3 closed sets that are all mutually congruent. Then all distances are realized within each set. Here a distance d is realized within a set S , if there are points x, y in S at distance d apart.

Covering a sphere with spheres

We work throughout in n -dimensional Euclidean space. It has been clear, since the publication of [1], that it should be possible to obtain quite good upper bounds for the number of spherical caps of chord 2 required to cover the surface of a sphere of radius R > 1, and for the number of spheres of radius 1 required to cover a sphere of radius R > 1. But it is not quite simple to organize the necessary calculations to give estimates which are manageable, and which are as good as the method allows for all R > 1. The following results seem to be a reasonable compromise between precision and simplicity; but, for reasons we will give later, they are not completely satisfactory.

Functions continuous and singular with respect to a Hausdorff measure

1. Introduction . Let I 0 be a closed rectangle in Euclidean n -space, and let ℬ be the field of Borel subsets of I 0 . Let ℱ be the space of completely additive set functions F , having a finite real value F ( E ) for each E of ℬ, and left undefined for sets E not in ℬ. In recent work, we used Hausdorff measures in an attempt to analyze the set functions F of ℱ. If h ( t ) is a monotonic increasing continuous function of t with h (0) = 0, a measure h-m ( E ) is generated by the method first defined by Hausdorff [2].

The existence of a centrally symmetric convex body with central sections that are unexpectedly small

Let K, K ′ be two centrally symmetric convex bodies in E n , with their centres at the origin o. Let V r 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 E n , 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.

Some extremal problems for convex bodies

If K is a convex body in n -dimensional space, let SK denote the closed n -dimensional sphere with centre at the origin and with volume equal to that of K . If H and K are two such convex bodies let C ( H , K ) denote the least convex cover of the union of H and K , and let V*(H, K) denote the maximum, taken over all points x for which the intersection is not empty, of the volume of the set . The object of this paper is to discuss some of the more interesting consequences of the following general theorem.

σ-fragmentable Banach spaces

§1. Introduction . Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X . The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ -diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each e>0, it is possible to write where each set X i , i ≥1, has the property that each non-empty subset of X i , has a non-empty relatively open subset of ρ -diameter less than e. If is any family of subsets of X , we say that X is σ-fragmented by the metric ρ , using sets from , if, for each e>0, the sets X i , i ≥ 1, in (1.1) can be taken from

Covering space with equal spheres

In a recent paper Rogers [13] has discussed packings of equal spheres in n -dimensional space and has shown that the density of such a packing cannot exceed a certain ratio σ n . In this paper, we discuss coverings of space with equal spheres and, by using a method which is in some respects dual to that used by Rogers, we show that the density of such a covering must always be at least

Borel isomorphisms at the first level—I

A Borel isomorphism that, together with its inverse, maps ℱ σ -sets to ℱ σ -sets will be said to be a Borel isomorphism at the first level. Such a Borel isomorphism will be called a first level isomorphism , for short. We study such first level isomorphisms between Polish spaces and between their Borel and analytic subsets.

Descriptive Borel Sets

The descriptive theory of Borel sets is developed for a fairly general class of spaces. For a satisfactory theory it seems to be necessary to work with a Hausdorff space subject to the condition that each open set can be expressed as a countable union of closed sets. Under this condition it is shown that the descriptive Borel sets form a Borel ring of analytic absolutely Borel sets containing the compact sets. It is shown that a set in a metric space is descriptive Borel if and only if it is Lindelöf and absolutely Borel.

Some Problems in the Geometry of Convex Bodies

In this note we discuss four problems. The first three remain totally intractable; the fourth has recently yielded some interesting results that are as yet incompletely understood.