L. Fejes Tóth

On the sum of distances determined by a pointset

where d denotes the diameter of the circumcircle of the points. The constant cannot be replaced by a greater one. In order to prove our theorem we denote the points by P~ . . . . ,P~,Fixing the points ~ . . . . , P,,, we consider S,, as a function of the point P~. Since the distances P~P., , . . . , P1P,~ are convex functions of P~, the same can b e stated of S,i. It follows that S,~ takes its maximum for a point P1 ly!ng on the boundary C of the circumcircle of the points. Therefore all points may be supposed to lie on C. Furthermore we suppose that C is a unit circle and that the cyclical order of the points is PI . . . . . P,,,. Introducing the notations P,+I = P1, . . . , P~,~ 1 P,,-~ we consider the sum

On the densest packing of convex discs

We shall give a simple new proof of the following known theorem [1, 2]. Theorem. The upper density of a packing of translates of a convex disc cannot exceed the density of the densest lattice-packing of these discs . In [] and [2] this theorem is proved for centrally symmetric discs. The general case can be reduced to this one by applying to the discs the known construction of central symmetrization. Our proof goes in a reverse way. We shall give a direct proof for a special family of asymmetric discs whose centrally symmetric images exhaust the family of centrally symmetric convex discs. Using the properties of the symmetrization, this implies the validity of the theorem for all centrally symmetric discs, and consequently for all convex discs. This procedure of going from a special case to the general one, by applying the symmetrization twice, is illustrated by the following example. The validity of the theorem for a Reuleaux triangle implies its validity for a circle, which implies its validity for any disc of constant width.

Minkowski Circle Packings on the Sphere

We consider n caps on the sphere such that none of them contains in its interior the center of another. We give an upper bound for the total area of the caps, which is sharp for n = 3 , 4, 6, and 12 and is asymptotically sharp for great values of n .Abstract. We consider n caps on the sphere such that none of them contains in its interior the center of another. We give an upper bound for the total area of the caps, which is sharp for n = 3 , 4, 6, and 12 and is asymptotically sharp for great values of n .

Packing and covering with convex discs

Before turning to the questions to be considered in this paper, we recall two other problems. Let C ( a, p ) be the class of all convex discs of area not less than a given constant a and perimeter not greater than a given constant p . What is the densest packing and what is the most economical covering of the Euclidean plane with discs from C ( a, p )? Both problems are interesting only if p 2 / a i.e . if p is less than the perimeter of a regular hexagon of area a . In this case, the densest packing arises from a regular hexagonal tiling by rounding off the corners of the tiles by equal circular arcs so as to obtain smooth hexagons of area a and perimeter p .

Elementarer Beweis einer isoperimetrischen Ungleichung

Es sei bemerk t , dass bei dem tJbergang yon einem nicht konvexen Gebiet zur konvexen Hiille L verkleinert , F vergrtissert wird, wiihrend R unveriinclert bleibt. Daher geniigt es die Ungleichung (2) fiir konv~xe Vielecke zu beweisen. Dagegen kann yon der Giihigkeit der Ungleichung (1) fiir konvexe Vielecke nicht unmi t t e lba r auf ihre Giihigkeit fiir beliebige Vielecke gefolgert werden. I m Folgenden werden beide Ungleichungen auf einen Schlag bewiesen und zwar fiir beliebige Vielecke. ~ Der Beweis beruh t auf zwei Hilfssiitzen :

SYMMETRY INDUCED BY ECONOMY

Abstract Various extremum problems are presented which lead to highly symmetric geometrical configurations.

A geometrical analogue of the phase transformation of crystals

Abstract A geometrical problem is discussed whose solution shows a close analogy to the transformation of one allotropic crystalline form to another.

Some Researches Inspired by H. S. M. Coxeter

Let me start with extremum properties of the regular solids, pointing out how fertile a remark of Professor Coxeter turned out to be in this field. The researches started thirty years ago by Coxeter’s remark are still in progress.

Flight in a packing of disks

For a packing in the plane consisting of open convex domains, none of which contains the originO, let?(r) be the length of the shortest path which evades the domains and connectsO with a point at a distancer fromO. Upper and lower bounds are given for the supremum of?(r) taken over all packings whose members are domains of diameter at least 2.

Densest packing of translates of the union of two circles

Letu be the union of two unit circles whose centers have a distance at most 2. Motivated by more general problems it is proved that the density of a packing of translates ofu never exceeds the density of the densest lattice-packing.