I need to learn some basics from convex vector optimization and this term (lower image) I somehow cannot grasp... here is context:
$ \max f(x) \qquad (\text{ with respect to the} \leq_K)\qquad (P)\\ \text{s.t.} \;\; g(x)\leq 0.$
$\cdot$ $\chi = \{x \in X: g(x)\leq 0 \}$ is convex
$\cdot$ $LI= \text{cl}(f(\chi)-K)$ is called lower image of $(P)$
($K \subseteq \mathbb{R}^q $ is solid, pointed, polyhedral convex ordering cone, $f: \mathbb{R}^n \to \mathbb{R}^q $ is $K$-concave, $g:\mathbb{R}^n \to \mathbb{R}^m$ is $\mathbb{R}^m _+$-convex)
In optimization the set $\chi$ is called the feasible set. Those are the points among which we are searching the optimal ones.
In general, if $A$ is a set contained in the domain of a function $f$, then $f(A)$ is defined to be $$f(A) = \{f(x):\ x\in A\}$$
The set $f(A)$ is called the image of $A$ by applying $f$.
The next thing that is relevant is the subtraction $A - B$ of two sets of vectors. $$A-B = \{x -y:\ x\in A\text{ and }y\in B\}$$
In your case $K$ is an ordering cone. You need to grok what happens to a set $A$ when you subtract the cone $K$. Start by looking at the case of one point $A=\{p\}$. In that case $A-K = -K + A$ looks like placing the code with the vertex at $p$ but pointing in the opposite direction as $K$, i.e. downwards.
For a general $A$, $A-K$ would look like a union of copies of $K$ starting at every point of $A$ and pointing in the opposite direction as $K$; like $A$ casting a shadow downwards.
In your case $A$ would be $F(\chi)$. So, it is like the downward shadow of the image of $\chi$ by $f$. I guess that is why they are calling it lower image.
Try simple examples: