I meet a problem in paper.
Assume $\delta\in(0,1)$ and $\mathbf z\neq\mathbf 0$. The author says that by applying the strict convexity of $f(\mathbf x)$, it is readily obtain $$f(\mathbf x+\delta\mathbf z)>f(\mathbf x)+\delta\nabla f(\mathbf x)^T\mathbf z$$
However, I think the inequality still holds if $f(\mathbf x)$ is only a convex function.
As we all know, a function is called a convex function if and only if $$f(\mathbf x)\geq f(\mathbf x_0)+f'(\mathbf x_0)(\mathbf x-\mathbf x_0)$$
I want to know whether there exists another point except $\mathbf x=\mathbf x_0$ such that equality holds.
Thanks a lot for any help.