Convex hull of $3$ dimensional set reduced to $2$ dimensional set Let $S = \{(f_1(t), f_2(t), f_3(t)) : t \in \mathbb{R}\}$ and suppose $f_3(t) \geq 1$ for all $t \in \mathbb{R}$. Is finding the convex hull of $S$ in some way equivalent to find the convex hull of $T = \{f_1(t)/f_3(t), f_2(t)/f_3(t)\}$?

In particular, $S = \{(a^3 + a, a^2, a^4 + 1) : a \in \mathbb{R}\}$ here.
Not at all. If $S$  is a $3-$dimensional it must be the subset of its convex hull. Then the dimension of Convex hull cannot be smaller than of the set itself.
Not at all. If $S$  is a $3-$dimensional it must be the subset of its convex hull. Then the dimension of Convex hull cannot be smaller than of the set itself.