The mapping \(\gamma_{x,y}\) in normed linear spaces and applications (Q1363590)

In an arbitrary normed space one can define upper and lower semi-inner products by: \[ (y,x)_i:=\lim_{t\to 0^-}(|x+ty|^2-|x|^2)/2t\text{ and } (y,x)_s:=\lim_{t\to 0^+}(|x+ty|^2-|x|^2)/2t. \] [see, for example, \textit{D. Amir}, ``Characterizations of inner product spaces'' (1986; Zbl 0617.46030)]. The authors begin with a long list of the properties of these functions. For each pair of linearly independent vectors \(x,y\) the authors define a function \(\gamma_{x,y}(t):=(|x+2ty|-|x+ty|)/t\) on \(\mathbb{R}\setminus\{0\}\). Among many properties, the authors show that \(\gamma_{x,y}\) is continuous and increasing; that \(|x|\lim_{t\to 0^-}\gamma_{x,y}(t)=(y,x)_i\); \(|x|\lim_{t\to 0^+}\gamma_{x,y}(t)=(y,x)_s\); and that \(\lim_{t\to \pm\infty}\gamma_{x,y}(t)=\pm|y|\). These relationships are used to show that \(x\) and \(y\) are orthogonal (in the Birkhoff sense) if and only if \(\gamma_{x,y}(u) \leq 0 \leq \gamma_{x,y}(t)\) for all \(u,t\) with \(u < 0 < t\). The function \(\gamma_{x,y}(t)\) is also of interest in inner-product spaces and there it is shown to be twice differentiable.