ROTUNDITY AND SMOOTHNESS OF CONVEX-BODIES IN REFLEXIVE AND NONREFLEXIVE SPACES

For combining two convex bodies C and D to produce a third body, two o f the most important ways are the operation -/+ of forming the closure of the vector sum C + D and the operation <(gamma)over bar> of formin g the closure of the convex hull of C boolean OR D. When the containin g normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable wit h respect to the operation -/+ and the class of all smooth bodies in X is stable with respect to both -/+ and <(gamma)over bar>. In our pape r it is shown that when X is separable, these stability properties of rotundity (resp. smoothness) are actually equivalent to the reflexivit y of X. The characterizations remain valid for each nonseparable X tha t contains a rotund (resp. smooth) body.