Anybody have any tips on how to show that the function $\frac{1}{2}\vert x_1 + x_2 + x_3 - x_1x_2x_3\vert^2$ is convex in $\mathbf{x}$, where $\vert x_i \vert \leq 1$?

This is the squared norm of the $\beta$ spinor of a length-3 RF pulse in NMR under a parameterization I have been working with which I hope leads to a convex optimization problem for RF pulse design.

It is of course straightforward to calculate the Hessian of this function, but it is not readily apparent to me that the Hessian is convex. A Monte Carlo simulation over the range of the function does not reveal any $\mathbf{x}$ for which the Hessian has negative eigenvalues. So I believe this N=3 function is convex. However, what I am hoping is to find a way to show that the squared norm of a similar function is convex for any N.

Thanks!