Equilateral, Diametral, Centered Sets and Subsets of Spheres

Let X be a Banach space. Set, for x ∈ X and r ≥ 0 U(x, r) = {y ∈ X : ||x− y|| = r}. Given a nonempty, bounded set A ⊂ X, we set r(A, x) = sup{||x−a|| : a ∈ A} (x ∈ X) (radius of A with respect to x); r(A) = inf{r(A; x) : x ∈ X} (radius of A); δ(A) = sup{||a− b|| : a, b ∈ A} (diameter of A). Clearly, δ(A) ≤ 2r(A) always. Define, for A, the following properties: A is diametral: r(A, x) = δ(A) for every x ∈ A; A is equilateral: there exists k ∈ R+ such that ||a− b|| = k for all a, b ∈ A; A lies on a sphere: there exist x ∈ X and α > 0 such that for all a ∈ A, ||x− a|| = α (that is, A ⊂ U(x, α)). Clearly, A ⊂ U(x, α) ⇔ A−x α ∈ U(θ, 1). A is centered (with respect to x): there exists an x ∈ X such that a− x ∈ A ⇒ x− a ∈ A. In other terms, A is centered if it is the translate of a symmetric set (A− x is symmetric with respect to the origin). It is simple to see that if A is centered and bounded, then the point x appearing in the definition is unique.