Intersections of Centred Sets in Normed Spaces

Various kinds of closed centred sets in normed spaces are considered. Necessary and sufficient conditions are obtained for every decreasing sequence of such sets to have nonempty intersection. Let X be a real or complex normed space. Recall that a set S ⊆ X is symmetric if S = −S and balanced if tS ⊆ S for all t ∈ [−1, 1]. It is convenient to introduce the following definitions. Definitions. 1. The set S ⊆ X is centred if (a) S is closed (in the norm topology), and (b) there exists x0 ∈ S, called a centre of S, such that S − x0 is symmetric. 2. The set S ⊆ X is a star if (a) S is closed, and (b) there exists x0 ∈ S such that S − x0 is balanced. (Note that every star is a centred set with centre x0.) Remarks. 1. A set is centred if and only if it is a translate of a closed symmetric set containing 0. 2. A set is a star if and only if it is a translate of a closed balanced set. 3. If x0 and x1 are distinct centres of the centred set S then S contains the unbounded set {x0 + n(x1 − x0) : n ∈ Z}. In particular, if S is bounded then S has a unique centre. 4. If S is a star (resp. centred set) in X with centre x0 and T : X → Y is an affine map from X into the normed space Y then T (S) is a star (resp. centred set) with centre T (x0). Natural examples of stars in normed spaces include all closed affine subspaces and all closed balls B(x0, r) = {x ∈ X : ‖x− x0‖ ≤ r}. The purpose of this note is to consider the following natural question. Question. Let (Sn) be a decreasing sequence (i.e., Sn+1 ⊆ Sn) of stars (resp. centred sets) in the normed space X. When can we conclude that ∩Sn 6= ∅? Our main result, Theorem 2 below, answers this question for bounded sets. As motivation let us recall two related results: the first is a well-known exercise and the second a theorem of S̆mulyan [4] (see also [3,p.433]). (R1) In a Banach space X every decreasing sequence of closed balls has nonempty intersection. 1991 Mathematics Subject Classification. 46B20.