Characterizations of hilbertian norms involving the areas of triangles in a smooth space

Abstract

In the previous paper, we have defined together with I. Ionic\u{a} the heights of a nontrivial triangle with respect to Birkhoff orthogonality in a real smooth space $X$, $\mbox{dim}\, X\geq 2.$ In the present paper, we remark that, generally, the area of a nontrivial triangle in $X$ has not the same value for different heights of the triangle. The purpose of this paper is to characterize the norm of $X$ if this space has the property that the area of any triangle is well defined (independent of considered height). In this line we give five equivalent properties using the directional derivative of the norm. If $X$ is strictly convex and $\mbox{dim} X\geq 3$, then each of these five properties characterizes the hilbertian norms (generated by inner products).