Theorems on the existence of separating surfaces

Abstract

LetR andG be finite of sets inEd. This paper presents theorems on the existence of strict linear and spherical separators ofR andG that are similar to the fundamental separation theorem of Kirchberger. Kirchberger's theorem impliet that the strict linear separability of finite setsR andG is determined by the separability of all subsets of up tod+2 points ofR?G. This paper shows that under certain conditions, the linear separability ofR andG is determined by the separability of significantly fewer than all subfamilies of up tod+2 members ofR ?G. The same treatment is made of Lay's extension of Kirchberger's theorem to separation by hyperspheres.