  Motion planning is one of the key modules in autonomous driving systems to
generate trajectories for self-driving vehicles to follow. A common motion
planning approach is to generate trajectories within semantic safe corridors.
The trajectories are generated by optimizing parametric curves (e.g. Bezier
curves) according to an objective function. To guarantee safety, the curves are
required to satisfy the convex hull property, and be contained within the
safety corridors. The convex hull property however does not necessary hold for
time-dependent corridors, and depends on the shape of corridors. The existing
approaches only support simple shape corridors, which is restrictive in
real-world, complex scenarios. In this paper, we provide a sufficient condition
for general convex, spatio-temporal corridors with theoretical proof of
guaranteed convex hull property. The theorem allows for using more complicated
shapes to generate spatio-temporal corridors and minimizing the uncovered
search space to $O(\frac{1}{n^2})$ compared to $O(1)$ of trapezoidal corridors,
which can improve the optimality of the solution. Simulation results show that
using general convex corridors yields less harsh brakes, hence improving the
overall smoothness of the resulting trajectories.
