For a convex shape $C(r, q)$, where $r$ represents the position of a point in the convex shape and $q$ represents the orientation using quaternions, can we show that the correspondence that maps from $(r, q)$ to the interior and surface of a convex shape is a continuous correspondence?
From my understanding $C(\cdot)$ is a correspondence instead of a function. Thus, we need to show that it is both upper and lower hemicontinuous in order for it to be continuous.
For upper hemicontinuous, we would need to show that for any open set $V \subset \mathbb{R}^3$ such that $C(r, q) \subseteq V$, there exists an open set $U$ where $(r, q) \in U$, where we have if $(\bar{r}, \bar{q}) \in U$ we have $C(\bar{x}, \bar{q}) \subseteq V $. I think this makes sense, for an infinitesimally small change between $(r, q)$ and $(\bar{r}, \bar{q})$ the convex shape should also change infinitesimally. But how would I show this?
Similarly, for lower hemicontinuous, we would need to show that for any open set $V \subset \mathbb{R}^3$ such that $C(r, q) \cap V$, there exists an open set $U$ where $(r, q) \in U$, where we have if $(\bar{r}, \bar{q}) \in U$ we have $C(\bar{x}, \bar{q}) \cap V $.
If this cannot be shown for general convex shapes, can it be shown for polygons and capsules?
Thanks in advance!