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There have been many contributions of work in different fields of convexity giving various metrical properties of convex sets. In this thesis we shall consider some further ideas which seem interesting to study. A standard way of tackling certain types of problems is to prove the existence of an 'extremal' convex set with respect to the property in consideration and by a series of arguments determine its construction. Generally speaking the extremal set turns out to be regular in some sense with a correspondingly easy geometry. In Chapters 1 and 2 we shall concern ourselves entirely with polytopes and we shall give some results on the metric properties of their faces. Following these results, we shall in Chapter 3 consider some continuity properties of the more general class of cell-complexes. In Chapters 4, 5 and 6, we shall confine ourselves to the plane. In Chapter 4, we shall consider sets which incertain senses correspond to the sets of constant width. This leads us in Chapter 5 to give some results concerning the minimal widths of triangles circumscribing convex sets. Finally, in Chapter 6 we consider the areas of certain subsets of a convex set which are determined by partitions of that set by three concurrent lines. Papers which are relevant to the field of study in a particular chapter are mentioned briefly in an introduction to that chapter.

The object of this thesis is to solve, in integers X and Y, various equations of the form [equation] where d and N are given square free integers. The work stems from two papers by J.H.E. Cohn in which the equations [equation] are solved for certain values of d. It is well known that the solutions of X2-dY2=4, and those of X2-dY2 = -4 where such solutions exist, may be expressed in terms of the least positive solutions of these equations. Solutions of the equations [equations] may now be sought among those of x2-dY2 = +1,+4. Extensive work in this direction has been done by W. Ljunggren working in the quadratic field R(d2) and other allied algebraic fields. His methods are powerful but deep and complicated. It is possible to show that the solutions of the equations X2 for sequences which satisfy a three-term recurrence relation. By applying the elementary theory of quadratic residues to these equations Cohn has solved the equations and for those d for which either of the equations has solutions (X,Y) for which X and Y are both odd. This thesis extends Cohn's work, Using similar methods, to solve the equations and for the same values of d as above and any given integer N. A few limited results are given for other values of d. L.J. Mordell has given simple conditions under which the equation can have no solutions. A theorem of a similar type concerning the equations x4-dY2 = 1,4 is proved. Finally the results proved in this thesis are compared with those of Cohn, Ljunggren and Mordell.

The problem we consider here arises quite naturally from Crum's Problem which asks: What is the maximum number of non-overlapping polyhedra such that any pair of them have a common boundary of positive area? In (1) Besicovitch, by constructing a sequence of polyhedra satisfying the required conditions, showed that the answer to Crum's Problem is infinity. In this thesis we ask: What is the answer when the polyhedra of Crum's Problem are restricted to be tetrahedra? As stated in the Abstract, we prove that the answer is either 8 or 9 and the evidence tends to point to the answer in fact being 8. As the difference in answers would suggest, the methods we use to establish these results are completely different from those used by Besicovitch in his paper. In Chapter One we show that an n-con (for definition see the Abstract) can be represented by an n-Towed matrix whose minors satisfy certain conditional we then develop arguments from which we deduce that n is less than 18. Chapter Two shows that the bound may be reduced to n less than 14. The subsequent five chapters are mainly concerned with the conditions under which a 9-con can exist and we eventually show that if an n-con for n > 9 exists then no plane contains six faces of the tetrahedra of the n-con.5.Our analysis of the 9-con continues in Chapter Eight where we show that what one may describe as the most symmetrical case for a 9-con cannot exist and also that the faces of the tetrahedra of the 9-con must be so arranged that they are contained in either nine or ten planes. To demonstrate that the existence or not of a 9-con is critical, we show in Chapter Nine that a 10-con cannot exist and in Chapter Ten that a 8-con does exist. Chapter Eleven discusses the results obtained.

In Chapter I we begin by considering a theorem of J. Dieudonne on the minimum radius of starlikeness of a class of analytic functions. We give a simple new proof of this theorem. By this new proof also we find the minimum radius of univalence of this class and we determine all the cases which give the minimum radius of univalence and the minimum radius of starlikeness. We then use a method similar to that in this new proof to obtain the minimum radius of univalence and the minimum radius of starlikeness of some other classes of analytic functions. For each class we determine all the cases giving the minimum radius of univalence and the minimum radius of starlikeness. Then we give some similar results for the minimum radius of convexity. In Chapter II first we deal with Heawood's Lemma which was established and used by P.J. Heawood to prove the theorem known as the Grace-Heawood Theorem. The same lemma was used by S. Kakeya in the proof of another theorem. We show that Heawood's Lemma is false and we give new proofs of these results. Then for some special cases we improve the value of the radius of univalencegiven by Kakeya's Theorem. In this connection we firstgive L.N. Cakalov's result and then we obtain some improvements of his result. In Appendix I we give some examples related with Chapter I and Chapter II. In Appendix II we give an example which shows that there is an error in a paper by M. Robertson.