Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc length be a convex function? Is there any definite relationship..
For the toroidal spiral curve parametrized by: $$x(t) = (a + b \cos(qt)) \cos (pt)$$ $$y(t) = (a + b \cos (qt)) \sin (pt)$$ $$z(t) = b \sin (qt) $$
is the arc-length function convex in t? Does it depend on the choice of a,b,p and q?
Given that an arbitrary regular curve $\gamma$ can be parametrized by arc length, there's no hope of deducing convexity of the arc length function $$ s(t) = \int_{a}^{t} \|\gamma'\| $$ from the shape of the curve. Convexity of the arc length function depends entirely on the parametrization.
In particular, convexity of a curve can have no bearing on convexity of the arc length function, just as Hagen von Eitzen noted.
If $p/q$ is rational (and both $p$ and $q$ are non-zero), the arc length of the toroidal spiral is neither concave nor convex because the speed is periodic and not constant. By a continuity argument, I believe you can make the same deduction for arbitrary non-zero $p$ and $q$. (If $p$ or $q$ is zero, the speed is obviously constant and the arc length is proportional to $t$.)