A New Look at Convexity and Pseudoconvexity

Convexity is a classical idea. Archimedes used a version of convexity in his considerations of arc length. Yet the idea was not formalized until 1934 in the monograph of Bonneson and Fenchel [BOF]. The classical definition of convexity is this: An open domain Ω ⊆ R is convex if, whenever P,Q ∈ Ω, then the segment PQ connecting P to Q lies in Ω. We call this the synthetic definition of convexity. It has the advantage of being elementary and accessible (see [VAL]). The disadvantages are that it is non-quantitative and non-analytic. It is of little use in situations of mathematical analysis where it is most likely to arise. The analytic definition of convexity is a bit more recondite. Let ΩßR have C boundary. For us this means that there exists a C function ρ defined in a neighborhood U of ∂Ω such that