And there are many such curves. Imagine a fixed length rod constrained to rotate around a fixed point but let the rod be free to slide towards any of its ends. Just trace a curve with one of the ends eventually bringing it to the initial location of the other end. With a jerky mouse movements it is hard to get a region that looks remotely convex. It should become apparent though that such regions do exist.
All convex regions are star-shaped.
In particular, triangles are star-shaped, as are quadrilaterals even if not convex.