Inspired by the question, "How to partition area of an ellipse into odd number of regions?," I ask for a partition an ellipse into three convex pieces, each of which has the same area and the same perimeter. The perimeter includes both arcs of the ellipse and whatever cuts are used.
This is known to be possible by recent results of Aronov and Hubbard, "Convex Equipartitions of volume and surface area," and by Karasev, "Equipartition of several measures," but perhaps the general techniques in these papers (which I have not studied) need not be used in this special case. Perhaps there is a natural construction?
Update. The two papers I cited above are both difficult for me to penetrate. The special case of equipartition into three parts was achieved earlier in a paper by Imre Bárány, Pavle Blagojevićc, and András Szűcsd, "Equipartitioning by a convex 3-fan," which I cannot access at the moment. But as you can infer from the title, the partition is accomplished via a convex 3-fan: a point with three rays emanating from that point.