D. G. Larman

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.

Convex bodies, economic cap coverings, random polytopes

Let K be a convex compact body with nonempty interior in the d-dimensional Euclidean space Rd and let x1, …, xn be random points in K, independently and uniformly distributed. Define Kn = conv {x1, …, xn}. Our main concern in this paper will be the behaviour of the deviation of vol Kn from vol K as a function of n, more precisely, the expectation of the random variable vol (K\Kn). We denote this expectation by E (K, n).

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.

A compact set of disjoint line segments in E 3 whose end set has positive measure

If L is a set of disjoint closed line segments in E n , let E ( L ) denote the end set of L , i.e. the set of end points of members of L . In [ 1, 2 ] V. L. Klee and M. Martin proved the following lemma: If L is a disjoint set of closed line segments in E 2 such that E ( L ) is compact, then E ( L ) has zero 2-measure.

A note on the false centre problem

If K is a set in n -dimensional Euclidean space E n , n ≥ 2, with non-empty interior, then a point p of E n 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. Hobinger [2] extended this theorem to any smooth convex body K witl a false centre p anywhere in E n . 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.

Projecting and uniformising Borel sets with σ, sections I.

Introduction . Suppose that M i s a set in a cartesian product space X × Y . If X and Y are both R 1 and M is a borel set, K. Kunugui [1] and W. J. Arsenin [2] have shown that those points which lie on sections {x} × Y which meet M in a nonempty K σ set project onto a set in X which is the complement of an analytic set. Somewhat later, W. J. Arsenin and A. A. Lyapunov [3] reproduced the proof from [2], still apparently without being aware of Kunuguis results. Although, as we shall see later, Kunuguis proof seems to be incorrect, it does contain an ingenious lemma (Lemma 5 below). Actually [2] also contains this lemma in a disguised form that seems, to me at least, to be less satisfactory. One of the purposes of this article is to give a promised proof, see D. G. Larman [4] of Kunuguis theorem in slightly more general circumstances by combining ideas of Kunugui [1] with those of P. Novikoflf [5].

On the selection of non-σ-finite subsets

Let A be a subset of a compact metric space Ω, and suppose that A has non-σ-finite h -measure, where h is some Hausdorff function. The following problem was suggested to me by Professor C. A. Rogers: If A is analytic, is it possible to construct 2 ℵo disjoint closed subsets of A which also have non -σ- finite h-measure? At this level of generality the problem, like others which involve selection of subsets, appears to offer some difficulty. Here we prove two results which were motivated by it.

The d – 2 Skeletons of polytopal approximations to a convex body in E d

Let K be a convex body in E d and let skel s K denote the s -skeleton of K . Let η s (K) denote the Hausdorff s -measure of skel s K and