I am wondering whether the following property, which seems intuitive in $\mathbb R$, holds in $n$-dimensional Euclidean space.
Question: Let a smooth, strongly convex function $V: \mathbb R^d \to \mathbb R$ (strongly convex meaning $\exists \alpha >0, \forall x \in \mathbb R^d: \nabla^2 V(x) \geq \alpha \operatorname{Id}$ as bilinear forms). Does there exists a diffeomorphism $\gamma : \mathbb R^d \to \mathbb R^d$ and a positive definite matrix $A$ such that the following property holds? \begin{equation*} \forall x \in \mathbb R^d: \gamma^{-1}\circ V\circ \gamma(x) = x^T A x \end{equation*}
Any references or help is appreciated.
Note: I am not interested in an expression for $\gamma$, just whether it exists.
Motivation: My motivation comes from convex optimisation, where it is easier to prove properties for quadratic optimisation problems than for arbitrary convex optimisation problems. Hence, I am wondering whether one can equate to any strongly convex function a strongly convex quadratic function up to a diffeomorphism.