Playing with the TikZ package for (La)TeX, I made a nice figure. Well, I think it is nice, anyway. You can ignore the distracting colours and the concentric circles, they are not important for this question. I would remove them, but I am not working at the computer where I made the picture.
The figure is obtained as follows: take a finite, nonempty sequence of points $p_1, \ldots, p_n$ in the plane (ideally the points are not all on the same line). Let $b$ be the barycentre. For each $i$ join the points $p_i$ and $p_{i + 1}$ with the quadratic Bezier curve whose "control point" is $b$. (Do the same for $p_n$ and $p_1$ to close the thing up, of course.)
Does this construction have a name? Is there anything interesting about it? Is there a generalisation to higher-degree Bezier curves or to higher dimensions? Of course we could see a degree 1 equivalent as being just a polygon.
I am afraid the javascript tag selector thing doesn't seem to be working for me here, so I took a guess at a suitable tag and left it at that. Maybe somebody will be able to fix this.
