Is there a way to describe any 2D shape from a set of independent parameters $p_0$, $p_1$, ... such that it is always convex no matter what the parameters are? The parameters can be limited to a fixed range $[{min}_i, {max}_i]$.
For example, you could say: $$x=p_0 \cos(\theta), y=p_1 \sin(\theta)$$ which would always give you a convex shape, but is limited to ellipses.
Or you could say: $$r=p_0 \sin(\theta + p_1) + p_2 \sin(2\theta + p_3) + ...$$
which should allow you to make any convex shape but it would also allow non-convex shapes.
Is there some parameterisation that ticks both boxes - allows all shapes (or at least a wide variety) but doesn't allow non-convex shapes?