Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and
$$
E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}.
$$


Then we have the following result which is



> 
> [Theorem:](https://math.stackexchange.com/a/4483272/1019043) If $X= \mathbb R^d$ and $f$ is convex, then the Hausdorff dimension of $E$ is at most $d-1$.
> 
> 
> 


Differentiability is a local property, so I guess above theorem is true even though $X \neq \mathbb R^d$. Can we extend above theorem to obtain below one?



> 
> If $X$ is convex and $f$ is convex, then the Hausdorff dimension of $E$ is at most $d-1$.
> 
> 
> 



Hausdorff dimension of the non-differentiability set a convex function
I just stumbled across your question. I have no idea how the proofs of these results go—and I am inclined to believe that they would indeed also prove the version you seek—but here's a way to deduce the local result from that on $\mathbf{R}^d$.


Let $X \subset \mathbf{R}^d$ be an open, convex set, $f: X \to \mathbf{R}$ be a convex function, and $E \subset X$ be the set of points where $f$ is not differentiable.


In general $f$ cannot be extended to a function defined on $\mathbf{R}^d$. *However, if it were bounded and Lipschitz, then it could be [Thm. 4.1,[Yan12](https://arxiv.org/pdf/1207.0944.pdf)].* Indeed, under these assumptions the extension $\bar{f}: \mathbf{R}^d \to \mathbf{R}$ can be defined by setting
\begin{equation}
\bar{f}(z) := \operatorname{sup} \{ t f(x) + (1-t) f(y) \mid x,y \in X, t \geq 1, z = t x + (1-t) y \}
\end{equation}
for all points $z \in \mathbf{R} \setminus X$.


Let $x \in X$, and $r > 0$ be so small that $D\_r(x) \subset \subset X$. Write $f\_{x,r}: D\_r(x) \to \mathbf{R}$ for the restriction of $f$ to this disk. As convex functions are locally Lipschitz, $f\_{x,r}$ is Lipschitz and bounded, and we may thus extend it to $\bar{f}\_{x,r} :\mathbf{R}^d \to \mathbf{R}$.


The global version gives that the Hausdorff dimension of $E \cap D\_r(x)$ is at most $d-1$. The conclusion follows after covering $X$ with a countable collection of such disks.


