Assume $X$ is a compact set (not necessarily convex). If the following equality holds true? $$\text{Conv}(X)=\text{Conv}(\text{extr(X)})$$ where $\text{extr}$ means the extreme point of $X$. The definition of the extreme point is that a point $a$ is said to be an extreme point of the set $X$ if the point $a$ cannot be represented as a convex combination of any other two distinct points in $X$.
Specifically, I am dealing with $X$ being a union of finite hyperrectangles ($X=\cup_{i=1}^{k}R_i$ where $R_i=\{r_{i,1},...,r_{i,m}:0\leq r_{i,j}\leq c_{i,j}\}$ for any $i\in\{1,...,k\}$ and $c_{i,j}$ may be different among indeces $i,j$) where each coordinate is non-negative.
I cannot find any counter-example that the equality is false.
Edit: I think that this equality is not true since one can easily construct three 2-D rectangles such that $\text{extr}(\text{Conv}(X))\subseteq\text{extr}(X)$, namely, $\text{Conv}(\text{extr}(\text{Conv}(X)))\subseteq\text{Conv}(\text{extr}(X))$. That is, we have $\text{Conv}(X)\subseteq\text{Conv}(\text{extr}(X))$.