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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.

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    I just deleted an "answer" in which I forgot to include the ray lengths in the perimeter definitions. THis problem is more difficult than I thought by far, and I now don't think it's easy to do if we select a point on the major axis of the ellipse at which to center the fan. At least it doesn't sweem to be a case of the one variable mean value theorem...2012-10-13

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Let $V_1 V_2$ be the major axis of our ellipse, with area $\Delta$. Take a point $P$ between $V_1$ and $V_2$ such that $PV_1=x$, and two points $Q_1,Q_2$ such that $Q_1 Q_2$ is perpendicular to $V_1 V_2$ and the elliptic sector $E_1$ delimited by the rays $PQ_1,PQ_2$ has area $\Delta/3$. Let $E_2$ be the elliptic sector delimited by the rays $PQ_1,PV_2$ and $p_j(x)$ the perimeter of $E_j$. The function

$ f(x) = p_1(x) - p_2(x) $

is clearly continous, so, if we find two points $x_0,y_0$ such that $f(x_0)f(y_0)<0$, we are sure that $f$ has at least a zero $z$ and we have done, since:

$ \Delta(E_2) = \frac{\Delta-\Delta(E_1)}{2} = \Delta(E_1). $

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    However, to ensure the convexity of $E_1$ we must have $x\geq 0.367534\dots V_1V_2$.2012-10-14