We show that every in nite dimensional Banach space has a closed and bounded convex set that is not remotal. This settles a problem raised by Sababheh and Khalil in [8].

In this paper, we study the Chebyshev centres of bounded subsets of normed spaces and obtain a norm inequality for relative centres. In particular, we prove that if T is a remotal subset of an inner product space H, and F is a star-shaped set at a relative Chebyshev centre c of T with respect to F, then llx - qT (x)1I2 2 Ilx-cll2 + Ilc-qT (c) 112 x E F, where qT : F + T is any choice functi...

In this paper, we prove that if E is an uniquely remotal subset of a real normed linear space such has Chebyshev center c ∈ and the farthest point map F : → restricted to [c, (c)] p...

in this paper, we study the chebyshev centres of bounded subsets of normed spaces and obtain a norm inequality for relative centres. in particular, we prove that if t is a remotal subset of an inner product space h, and f is a star-shaped set at a relative chebyshev centre c of t with respect to f, then llx - qt (x)1i2 2 ilx-cll2 + ilc-qt (c) 112 x e f, where qt : f + t is any choice function s...

This derivation of (2) is not valid, because this duality applies to inferences and theorems only, not to definitions. In fact, (1) should be treated as an axiom. Accordingly, the duality of Lemma 3.3.1 would not be valid until the dual (2) of (1) is proved independently, as we do below. Our new proof will make use of Proposition 3.3.3, the original proof of which is based on duality. So, we mu...

Answering a recent question posed by Gregori, Morillas and Sapena (“On a class of completable fuzzy metric spaces”, Fuzzy Sets and Systems, 161 (2010), 2193–2205) we present two examples of non strong fuzzy metrics (in the sense of George and Veeramani).

We provide an extension of the notion of chain-valued frame introduced by Pultr and Rodabaugh in [Category theoretic aspects of chain-valued frames: parts I and II, Fuzzy Sets and Systems 159 (2008) 501–528 and 529–558] by relaxing the assumption that L be a complete chain. As a result of this investigation we formulate the category L-Frm of L-frames under the weaker assumption that L is a comp...