Given the \(\alpha\)-convex functional \(\varphi\) on the Banach space \((X,\|\cdot\|)\), the authors consider the solution \(t\mapsto u(t)\) of the gradient flow of \(\varphi\) fulfilling a so-called evolution variational inequality of the form \[ {1\over 2} e^{\alpha t}\| u(t)- z\|^2-{1\over 2} e^{\alpha s}\| u(s)-z\|^2\leq(\varphi(z)- \varphi(u(t))) \int^t_s e^{\alpha r}dr, \] \[ \forall z\in X,\quad\forall 0< s< t< T. \] The main result of the paper asserts that, whenever a suitably large of solutions to the latter evolution variational inequality exist, the space \(X\) has to be necessarily a Hilbert space. This derivation is then used in order to discuss possible characterizations of Hilbert spaces based on quadrant flow evolution.