I'm new on stackexchange, thanks for this fantastic platform!
Let's get to the point. Suppose we have a function $I(x,y) \in \mathbb{R}$, $I(x,y) \geq 0$, convex, and differentiable for $x \in X$ compact subset of $\mathbb{R}$, and $y \in \mathbb{R}$. Is
$$I(\omega) := \inf_{x \in X \\ {y = \omega x}} I(x,y) ,$$
$\omega \in \mathbb{R}$, still differentiable? Please, note that the function defined as $\tilde{I}(x,\omega) := I(x,\omega x)$ in general is not convex anymore.