Let $(X, \mathcal{T})$ and $(Y, \mathcal{T}')$ be two topological spaces.
We call a correspondence (set-valued function) $F: X \rightarrow 2^Y$ lower semicontinuous if for every $G \in \mathcal{T}'$ , $\{x \in X\mid F(x) \cap G \neq \phi\} \in \mathcal{T}$ holds.
Consider a lower semi-continuous correspondence $F: X \rightarrow 2^Y: x \mapsto F(x)$. Show that $c(F): X \rightarrow 2^Y: x \mapsto c(F(x))$ , where $c(A)$ denotes the convex hull of $A$ in $Y$, is also lower semi-continuous.
This is a claim I came across in C.D. Aliprantis and K.C. Border's "Infinite Dimensional Analysis: A Hitchhiker's Guide". It is also a proposition in Ernest Michael's first paper on Continuous selections. However, I'm failing to prove this by just using the definitions I know.
We will consider partial case : Assume that $X,\ Y$ are isometric to $\mathbb{R}^2$.
For a fixed $G\in \mathcal{T}'$ define $B= \bigg\{ x\in X \bigg| c(F(x))\bigcap G \neq \emptyset \bigg\}$
Let $g\in c(F(x)) \bigcap G\neq \emptyset$ Hence $x\in B$ We will show that there is open set $T$ with $x\in T$ and $T\subset B$ :
Hence $g =\sum_{i=1}^3\ c_iy_i $ is a convex combination of $y_i,\ 1\leq i\leq 3$ in $F(x)$. And assume that open ball $B(g,r)$ is in $G$.
Consider open balls $B(y_i,r_i)$ Define $$ T =\bigcap_{i=1}^3\ \bigg\{ a\bigg|F(a)\bigcap B(y_i,r_i)\neq \emptyset\bigg\}$$ Here $T$ is intersection of three open sets, since $F$ is semi-continuous And $x\in T$ so that $T$ is nonempty open set.
Assume that $ \sum_{i=1}^3\ c_i B(y_i,r_i)$ is in $B(g,r)$ for sufficiently small $r_i$. Hence $B$ contains an open set $T$