Covering problems in euclidean space

Abstract

Our purpose in these pages will be to develop a broad survey of some problems in covering which have been solved using the methods of convexity. The basic aim has been to show the flexibility of approach which is desirable, and to do this we shall discuss ten particular problems with widely differing methods of solution. The problems to be discussed fall into three main categories. covering a single set by a single set with specified properties, covering a single set by many sets, and covering many sets by a single set. The first chapter is a collection of some necessary preliminary definitions and theorems in convexity. In the second chapter we consider six particular problems of covering a single set by a single set: these include finding the convex covers of certain arcs in two and three dimensions, finding the smallest containing sphere of an n-dimensional set, and establishing properties of particular types of containing set. The final chapter contains problems which fall into the other two categories outlined above, and includes finding a single set to cover an infinite class of sets, and covering one set by a finite class of sets. The last paper considered relates one set (the plane) and a given infinite class of sets: it is included for the sake of completeness and because of the methods of solution involved, but in fact can be considered as a problem in covering only if the latter term is not restricted to mean complete covering by a class of sets. A full list of references to the original papers is given at the end.