Define a convex shape $C(r, q)$, where $r$ represents the position of the convex shape and $q$ represents the orientation of the convex shape. Let $V$ be an open set in $\mathbb{R}^3$ where $C \subseteq V$. Can we show that there exists an open set $U$, with $(r, q) \in U$, such that $C(\bar{r}, \bar{q}) \subseteq V$ for any $(\bar{r}, \bar{q}) \in U$.
The type of convex shape I have in mind are like polygons and ellipsoids, and $C(r, q)$ is the convex set that is both the interior and the surface of the convex shapes in mind. These convex sets are compact, which I think from the Heince-Borel Theorem implies that they are also closed and bounded.
Thank you for the help!
Suppose the compact set $C$ is a subset of the open set $V$. Then each point of $c$ has an open neighborhood in $V$. Compactness implies that a finite subset of those neighborhoods covers $C$. Then a small change in position stays in that finite cover.