THEOREMS ON THE EXISTENCE OF SEPARATING SURFACES

Let R and G be finite of sets in E d. This paper presents theorems on the existence of strict linear and spherical separators of R and G that are similar to the fundamental separation theorem of Kirchberger. Kirchberger's theorem impliet that the strict linear separability of finite sets R and G is determined by the separability of all subsets of up to d+2 points of R⊃G. This paper shows that under certain conditions, the linear separability of R and G is determined by the separability of significantly fewer than all subfamilies of up to d+2 members of R ⊃G. The same treatment is made of Lay's extension of Kirchberger's theorem to separation by hyperspheres. © 1991 Springer-Verlag New York Inc.