Convexity Through the Ages

Abstract

In his treatise “On the sphere and cylinder” Archimedes defines a convex arc as a plane curve which lies on one side of the line joining its endpoints and all chords of which lie on the same side of it. Analogously he defines a convex surface bounded by a plane curve. His determination of arc lengths is based on certain postulates. One of these is: If one of two convex arcs with common endpoints lies between the other and the line joining the endpoints, the length of the first arc is smaller than that of the second. The determination of surface areas is founded on an analogous postulate.

Cependent, les théories ont leurs commencements: des allusions vagues, des essais inachevés, des problèmes particuliers; et même lorsque ces commencements importent peu dans l’état actuel de la Science, on aurait tort de les passer sous silence.

F. Riesz, 1913.

The following is essentially a translation of a talk given at the celebration of the centenary of the Danish Mathematical Society in 1973. Readers are kindly requested to take into consideration that the audience did not consist of experts in convexity. (The original version appeared in “Dansk Matematisk Forening 1923–1973”, Copenhagen 1973, p. 103–116.)