Nested sequences of stars and starshaped sets

In this paper we consider nested (decreasing and increasing) sequences of stars or starshaped sets in Banach spaces. The intersection, if decreasing, and the closure of the union, if increasing, are studied with regards to the preservation of these properties. Among other results we show that the closure of an increasing sequence of stars is a star if and only if the sequence of their centers is weakly convergent. Similar results, for starshaped sets, are true exactly in reflexive spaces.