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