Let $S \subseteq \mathbb{R}^n$ be a nonempty set and $p \in S$ be a point. A (closed) halfspace $H\subseteq \mathbb{R}^n$ is said to support $S$ at $p$ if $S \subseteq H$ and $p \in \partial H$. For each $p \in S$, let $$\mathrm{sh}_p(S):= \left\{\mbox{halfspace }H: \mbox{$H$ supports $S$ at $p$}\right\}.$$
Now, suppose the following assumptions: 1. $S$ is closed and 2. for every $p \in S$, $\mathrm{sh}_p(S)$ is nonempty.
My question is: is the following true? $$ \mathrm{conv.hull}(S) = \bigcap_{p\in S}\bigcap_{H \in \mathrm{sh}_p(S)} H $$
I agree with Zim‘s comment that there is an issue with closedness, and that if you would als about the closure of the convex hull, the statement would be true. Further, the statement would also be true if S is compact.
The way it is stated now though, the statement seems to be false:
Let S be the union of the x-axis and the point (0,1). For each point in S the set of supporting halfplanes is not empty. For every point on the d-axis, it consists of the closes halfplane above the x-axis, and for the point (0,1) it consists of the halfplane below the line y=1. In particular, the right hand side of your statement is the closed strip between the x-axis and the line y=1. However, the convex hull of S does not include the line y=1.