interchange of infinite intersection and taking convex hull of a set

Question

Let $B(x,\delta)$ be an open ball centered at $x\in R^n$ with radius $\delta>0$. Let $F:R^n\rightarrow R^m$ be a vector-valued function. Then $F(B(x,\delta))$ would be a subset of $R^m$. Let $\overline{co}\{A\}$ be the convex hull of set $A$. My question is the following: under what conditions, the following relation is true

\begin{equation} \bigcap_{\delta>0}\overline{co}\{F(B(x,\delta))\}=\overline{co}\bigg\{\bigcap_{\delta>0}F(B(x,\delta))\bigg\} \end{equation}

Note that $x$ is a discontinuous point of $F$

Answer 1

Just to show that $F:R^1\to R^1$ can be fairly nice without your relation holding, let $$F(x)=\begin{cases}1&\text{if }x=0,\\ x&\text{otherwise}. \end{cases}$$ Then $$\overline{\text{co}}\left\{\bigcap_{\delta>0}F(B(0,\delta))\right\}=\overline{\text{co}}\{1\}=\{1\}\subsetneq [0,1]=\bigcap_{\delta>0}\overline{\text{co}}\{F(B(0,\delta))\}.$$