Point processes in general position

The aim of this note is to introduce for point processes in ℝ d the notions general position and reinforced general position, and to characterize these processes. As a consequence we show that Poisson processes P ρ with an infinite intensity measures ρ are in general position iff ρ is diffuse in the sense that any affine subspace of dimension d − 1 is a ρ-nullset. Moreover, P ρ is in reinforced general position iff in addition any (d − 1)-sphere is a ρ-nullset.