A characterization of inner product spaces

Abstract: In this paper we present a new criterion on characterization of real inner product spaces. We conclude that a real normed space $(X, |...|)$ is an inner product space if $$\sum_{\epsilon_i \in {-1,1}} |x_1 + \sum_{i=2}^k\epsilon_ix_i|^2=\sum_{\epsilon_i \in {-1,1}} (|x_1| + \sum_{i=2}^k\epsilon_i|x_i|)^2,$$ for some positive integer $k\geq 2$ and all $x_1, ..., x_k \in X$. Conversely, if $(X, |...|)$ is an inner product space, then the equality above holds for all $k\geq 2$ and all $x_1, ..., x_k \in X$.