More characterizations of inner product spaces

Abstract

Let X be a real inner product space of dinmension at least three and let M be a 2-dimensional subspace of X. For a vector u in X but not in M, let v be the vector in M closest to u. It is easily seen that (i) if v = 0, then all of the vectors of norm 1 in M are equidistant from u and (ii) if v #0 and w = j v j -v, then of all the vectors of norm 1 in M w is the closest to u. The purpose of this paper is to show that each of these properties characterize those normed linear spaces which are inner product spaces. (For a survey of such results, see [3, pp. 115121].) Throughout, we let E denote real Euclidean 3-space. Our proofs are based on the following two characterizations of ellipsoids in E. Theorem A is due to G. Birkhoff [1]. Theorem B is due to Marchaud [4] and generalizes a result due to Blaschke [2, pp. 157-159].