On neighborly families of convex bodies

A family of convex bodies in Ed is called neighborly if the intersection of every two of them is (d - 1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in Ed, d ≥ 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in Ed such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes. © Birkhäuser Verlag, Basel, 2004.