Let $V$ be a finite-dimensional real vector space, let $P, Q \subseteq V$ be polytopes with $P \subseteq Q$. (Let a polytope be defined as the convex hull of finitely many points.) I'm interested in the following
Question: How can I identify the set of all affine maps $T: V \rightarrow V$ with $T(P) \subseteq Q$?
More precisely, I want to find a "nice" way to characterize the set of all maps $T: V \rightarrow V$ with the following properties:
- $T(\alpha v_1 + (1-\alpha) v_2) = \alpha T(v_1) + (1-\alpha) T(v_2) \quad \forall \alpha \in \mathbb{R}, \forall v_1, v_2 \in V$,
- $v \in P \Rightarrow T(v) \in Q$.
Admittedly, the question of how to find a "nice" way to characterize this set is rather vague. My motivation for this question is that I have given two such polytopes and want to study this set of maps, and it seems to me that this set is generic enough that people might have studied it.