A generalization of Kung's theorem

The theorem of Jung establishes a relation between circumradius and diameter of a convex body. The half of the diameter can be interpreted as the maximum of circumradii of all 1-dimensional sections or 1-dimensional orthogonal projections of a convex body. This point of view leads to two series of j-dimensional circumradii, defined via sections or projections. In this paper we study some relations between these circumradii and by this we find a natural generalization of Jung’s theorem. Introduction Throughout this paper E denotes the d-dimensional euclidean space and the set of all convex bodies K⊂Ed — compact convex sets — is denoted by Kd. The affine (convex) hull of a subset P ⊂ E is denoted by aff(P ) (conv(P )) and dim(P ) denotes the dimension of the affine hull of P . The interior of P is denoted by int(P ) and relint(P ) denotes the interior with respect to the affine hull of P . ‖ · ‖ denotes the euclidean norm and the set of all i-dimensional linear subspaces of E is denoted by Li . L denotes for L ∈ Li the orthogonal complement and for K ∈ Kd, L ∈ Li the orthogonal projection of K onto L is denoted by K|L. The diameter, circumradius and inradius of a convex body K ∈ Kd is denoted by D(K), R(K) and r(K), respectively. For a detailed description of these functionals we refer to the book [BF]. With this notation we can define the following i-dimensional circumradii Definition. For K ∈ Kd and 1 ≤ i ≤ d let i) R σ(K) := max L∈L i max x∈L R(K ∩ (x+ L)), ii) R π(K) := max L∈L i R(K|L). We obviously have R σ (K) ≥ R σ(K), R π (K) ≥ R π(K), R π(K) ≥ R σ(K) and R σ(K) = R d π(K) = R(K), R 1 σ(K) = R 1 π(K) = D(K)/2. The theorem of Jung [J] states a relation between the circumradius and the diameter of a convex body. On account of the definition of R σ(K), R 1 σ(K) we can describe his result as follows 1991 Mathematics Subject Classification. AMS 52A43.