ISOSCELES ORTHOGONAL TRIPLES IN LINEAR 2-NORMED SPACES

A triple (x, y, z) in a linear 2-normed space (X, parallel-to.,. parallel-to) is called an isosceles orthogonal triple, denoted \(x, y, z), if parallel-to x + y, z parallel-to = parallel-to x - y, z parallel-to, parallel-to x + z, y parallel-to = parallel-to x - z, y parallel-to, and parallel-to y + z, x parallel-to = parallel-to y - z, x parallel-to. \(.,.,.) is said to be homogeneous if \(x, y, z) implies \(ax, y, z) for all real a and it is additive if \(x1, y, z) and \(x2, y, z) imply that \(x1 + x2, y, z). In addition to developing some basic properties of \(.,.,.), this paper shows that under the assumption of strict convexity, every subspace of X of dimension less-than-or-equal-to 3 contains an isosceles orthogonal triple. Further, if (X, parallel-to.,.parallel-to) is strictly convex and \(.,.,.) is either homogeneous or additive, then (X, parallel-to.,.parallel-to) is a 2-inner product space.