a norm inequality for chebyshev centres

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 sending x to the point qt (x) with ilx - qt (x)11= supfet ilx - dl (note that t is called remotal if such a choice function qt exists). we then use such an inequality to show that, under some restrictions, a uniquely remotal set is a singleton. further, we show that if c is a centre of a remotal subset t of a norrned space e and x e e, then there exists a. functional f e e* such that i i f i1 i 1 and ilx - qt (x)1i2 l i i c - q*(c)112 + 2 if (x - c) 12 - iix - ~11