On a common generalization of Borsuk's and Radon's theorem

1. The well-known theorem of RADON [3] says that if A c R a and IAl=>d+2, then there exist B, C c A , B ( ~ C = ~ such that conv BNconv C is not empty. i t is clear that for each finite set A = { a l , ..., a,} in R e with n ~ d + 2 one can find a linear map f : Ra+I-*R a and a set A ' = {a; . . . . , a~}cR a+l such that f ( a ; )=a l i = l , 2, . . . , n and int conv A' is not empty and vert conv A ' = A ' . In view of this fact, Radon ' s theorem can be stated in the following way.