Geometric extremum problems were already investigated in ancient times, and the isoperimetric problem in particular attracted the attention of mathematicians. In the 19th century considerable progress was made by the work of Steiner, Lindelof, Schwarz and Brunn, among others. Minkowski's research on convex bodies and the application of geometric methods to number-theoretic questions gave an enormous impetus to the field of geometric extremum problems. In the present century a great number of very important results were obtained by Blaschke, Steinitz, Hadwiger and Dinghas, among many others. Beginning in the forties, the fruitful ideas and the contributions of L. Fejes Toth formed and advanced this area of geometry in a singular way. His books (L. Fejes Toth 1972, 1964), to which we shall refer continually, turned out to be an abundant source of inspiration to workers in this field. In this chapter we shall be dealing with extremum problems for convex sets mainly in 2- and 3-dimensional Euclidean space. Limitations of space force us to make a selection of the problems to be considered. We shall give a survey of the results, including the pertinent references, but only occasionally indicate the methods of proof.

Extremum Problems for Convex Discs and Polyhedra

Some extremal properties of convex sets

Kantenlängensumme, mittlere Breite und Umkugelradius konvexer Polytope

Man kennt heute eine Reihe von Extremaleigenschaften, welche einige oder alle regularen Polyeder des dreidimensionalen euklidischen Raumes E3 innerhalb bestimmter Polyederklassen charakterisieren. Dabei nimmt jeweils eine gewisse, mit den Polyedern verbundene Grose wie Volumen, Oberflache, Kantenkrummung oder Kantenlangensumme fur einige oder alle regularen Polyeder einen extremalen Wert an. Im Zusammenhang mit Verteilungsproblemen von Punkten auf der Kugeloberflache treten auch einige halbregulare Polyeder als Extremalkorper hervor. Viele Ergebnisse in dieser Richtung findet man in den bekannten Buchern [11] und [12] von Fejes Toth. Es soll hier ein Uberblick uber einige neuere Resultate auf diesem Gebiet gegeben werden.

Neuere Entwicklungen über reguläre Polyeder

In this paper all sets considered are assumed to be compact subsets of Euclidean Space E n . A number of results concerning the total edge-lengths of polyhedra have been given by various authors, many of which are mentioned in references in [1]. In [1], it was conjectured that all polytopes inscribed in the unit sphere and containing its centre have total edge-length greater than 2 n . This was proved true for simplicial polytopes and shown to be best possible in the sense that there exist simplices with the stated property and with total edge-length arbitrarily close to 2 n . In this paper we shall show that the bound is not always best possible if the magnitudes of the faces of such polytopes are restricted and we shall also give some related results on surface areas. This work was carried out while the author was a research student at Royal Holloway College, London and is a revised version of part of the author's thesis approved for the Ph.D. degree.

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On polytopes with small faces

All sets considered in this paper will be subsets of n -dimensional Eucidean space E n . In this paper, we shall consider the ‘total edge-lengths’ of polytopes which are inscribed in a given sphere and which contain its centre. We first, however, mention some related problems considered by various authors which may be of interest to the reader.

J. N. Lillington

Papers

A conjecture for polytopes