I defined a function $f_d(\mathbf{x}) = -\langle\frac{\mathbf{x}}{|\mathbf{x}|},d\rangle,\,\,\mathbf{x}\in\Omega_d$, where $\,\Omega_d=\{\mathbf{x}\in R^n|\,\langle\mathbf{x},d\rangle\ge0\}$, and $d\in R^n$ is a unit direction. It is easy to see that $\Omega_d$ is a convex set.
Also, $f_d(\mathbf{x})$ is simply $-\cos\theta_d$, where $\theta_d$ refers to the angle between $d$ and $\mathbf{x}$. Because $-\cos\theta$ is a strictly convex function in the interval $\theta\in [-\pi/2,\pi/2]$, I think $f_d(\mathbf{x})$ is also a convex function.
However, the problem is that the Bregman divergence requires the function $f$ to be a strictly convex function, but this function $f_d(\mathbf{x})$ is only a convex function. The strictness is not satisfied when some $\mathbf{x}_1, \mathbf{x}_2\in\Omega_d$ are linearly dependent.
In this case, can I claim that there exists a Bregman Divergence $D_f$ associated with the $f$ above?
If not, is there any other function of cosine similarity that satisfies the properties of Bregman Divergence?
Thank you.