Let x(t) be a periodic function on [0, 2$\pi$]. I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left( t\right) \right\} $ is star-convex with respect to the origin. Is this a problem with a known solution? Is there any literature on this problem?
My starting point is to see where such a parametric curve would "graze" a ray from the origin. The resulting trig equation is $x\left( t\right) \ddot{x}\left( t\right) -\dot{x}\left( t\right) ^{2}=0$. I can find the multiplicity-2 (or higher) zeroes of this trig equation and this way I would have implicit hypersurfaces in the space of the Fourier coefficients that would "carve out" regions of star-convex shapes.
