Take a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. Choose an arbitrary direction $d \in \mathbb{R}^n$ and consider the restriction of $f$ to the line through $x \in \mathbb{R}^n$ in the direction $d$:
$$f_d(t) := f(x + dt), \textrm{ } t \in \mathbb{R}.$$
My question is: What is $\partial f_d(0)$? It is easy to show that
$$d^T \partial f(x) = \{d^Tg : g \in \partial f(x)\} \subseteq \partial f_d(0).$$
It would be very pleasing to have equality here.
By the way, I am aware that this result is just a special case of the following:
For $f$ convex, $A$ some matrix and $b$ some vector, the function $g$ defined by $g(x) = f(Ax)$ is convex and $\partial g(x) = A^T \partial f(Ax)$ for all $x$.
But I'm looking for a proof that doesn't use this fact. In fact, I'm trying to prove the above fact!