I am indebted to Dr L S Bosanquet and Dr R G Cooke for advice and encouragement during the preparation of this note.

This paper forms part of a thesis approved for the Degree of Ph.D. in the University of London.

Many results in the geometry of numbers assert, in effect, that inequalities of a certain type are soluble in integers, the constant on the right of the inequality being the best possible. Recent work of Mahler often enables one to prove that such an inequality has infinitely many solutions. In this paper we develop the theory of inequalities with infinitely many solutions, and investigate more deeply some of the questions which naturally arise.

Absolute convergence is equivalent to unconditional convergence of series of points of a Banach space if and only if the space is finite-dimensional.

In collaboration with Geoffrey Shephard and James Taylor during that period his interest in convex geometry and Hausdorff Measure Theory widened. In particular, with Geoffrey Shephard, he produced sharp bounds for the volume of a difference body, a problem which had been open for 30 years.

Visiting Jo and Ambrose in Gray Close, Hampstead, was always amazing. Ambrose had the largest desk piled high with totally incomprehensible research papers. Towards the end of his life he had to have a second desk which quickly became covered too. At the last Birthday party I attended there, the room was full of mathematical colleagues who might have been on a different planet. Over these occasions my sister presided invariably with a cold collage for lunch or a polite tea. Domestic considerations were never high on their agenda.

Professor Rogers has written this economical and logical exposition of the theory of packing and covering at a time when the simplest general results are known and future progress seems likely to depend on detailed and complicated technical developments. The book treats mainly problems in $n$-dimensional space, where $n$ is larger than 3. The approach is quantitative and many estimates for packing and covering densities are obtained. The introduction gives a historical outline of the subject, stating results without proof, and the succeeding chapters contain a systematic account of the general results and their derivation. Some of the results have immediate applications in the theory of numbers, in analysis and in other branches of mathematics, while the quantitative approach may well prove to be of increasing importance for further developments.

This monograph, based on the work of the author and his collaborators, gives a very elegant and readable account of many important results in the theory of packings and coverings, lattice as well as non-lattice. The author has managed in most cases to include proofs of best known results without affecting the readability of this book, which is entirely self-contained.

This little book maintains the high standard of scholarly exposition which has distinguished the Cambridge Tracts for several generations.

The style is brisk, clear and interesting, without any fuss, and I have no doubt at all that the book will be found a most valuable source of information in a very readable setting.

This beautifully written and beautifully printed little book contains an astonishing amount of information. ... this book is a combination, unique as far as the reviewer knows, of pedagogy and description of an active if special field of analysis as it now exists. A bright student can pick up this book and read it by himself. When he has finished the book, he will be able to ask questions of his own and will also have a useful guide to the literature. Mature mathematicians who need facts about Hausdorff measures for their own purposes - surface theory, harmonic analysis, and so on - will find clear statements, clear proofs, and abundant references for further pursuit.

For sometime we have thought that the theory of measurable selectors was somewhat lacking, in that one knows little about the topological properties of measurable functions. It is surprising that in very general circumstances upper semi-continuous set-valued maps do have selectors that, although not continuous, are of the first Baire class; that is, are point-wise limits of sequences of continuous functions. In the book we are mainly concerned with proving results showing the existence of selections of the first Baire class. We give a number of geometrically interesting examples, and some unexpected consequences for functional analysis.

In the period after the Second World War, Rogers rapidly emerged in the forefront of the renaissance of the geometry of numbers. His work on the lattice constants of cylinders both convex and non-convex, on the reducibility of star bodies and on the implications for the existence of infinitely many lattice points in automorphic bodies, and on the successive minima of general sets with respect to a lattice remains definitive. Since about 1958, Professor Rogers' main research interests moved to the theory of Hausdorff measures, of analytic sets and of general convex bodies, to all of which he has made important contributions.

I will discuss the elementary concepts of Length, Area and Volume from an advanced standpoint. Although my readers will have acquired a clear intuitive understanding of these basic geometric quantities, they may not realise that these quantities are very far from elementary. Many deep investigations have been necessary to refine the concepts and many basic problems remain with only inadequate answers.