On parallelogram areas in normed linear spaces
The author offers characterizations of inner product spaces in terms of equations satisfied by areas of parallellograms defined among others in the following different ways: \[ \begin{aligned} A_1(x,y)&:=(\| x\|^2\| y|^2-r(x,y)r(y,x)\,)^{1/2},\qquad r(x,y):=\lim_{t\to+0}(\| x+ty\|^2-\| x\|^2)/(2t); \\ A_2(x,y)&:=(\|\:(x-y)y+\,(x-y)^2(\| y\|^2-r(x,y)\,)\| x-y\|^{-2}\: \|; \\ A_3(x,y)&:=\| x-y\|(\,\| x\|^2-\| x-y\|(1-(\| y\|^2-r(x,y)\,)\| x-y\|^{-2}\,)\,)^{1/2}.\end{aligned} \] One result offered is that a real normed linear space \((E,\|\;\|)\;(\dim E\geq 2)\;\) is an inner product space iff (at least) one (thus all) of the \(A_j\)'s satisfies \(A_j(z,x+y)^2+A_j(z,x-y)^2=2A_j(z,x)^2+2A_j(z,y)^2\quad(j=1,2,3)\) for all \(x,y,z\) in \(E.\)
scientific article; zbMATH DE number 5126567