Is ellipse a quadrature domain for arc-length measure? More precisely does there exist points $z_1,\cdots,z_n$ inside an ellipse $E$ and non zero constants $c_1,\cdots,c_n$ such that $$\int _Ep(z)\,ds_z=\sum_1^nc_kp(z_k)$$ holds for all polynomials $p$?

Motivation: In the literature, there are various results regarding characterization of domains in $\mathbb R^N$ via eigenvalues of integral (in particular potential) operators. A prime example is the following theorem of E. Fraenkel:

Theorem.(Fraenkel 2000) Let $G⊂\mathbb R^N$ be a bounded open set and let $ω_N$ be the surface measure of the unit-sphere in $\mathbb R^N$. Consider $$u(x)=\begin{cases} \frac{1}{2\pi}\int_G\log|x-y|dy, & n=2 \\ \frac{1}{N-2}\int_G\frac{1}{\|x-y\|^N-2}dy, & n\geq 3 \end{cases}.$$ If $u$ is constant on $∂G$ then $G$ is a ball.

One can ask similar question for the case that the operator on hand is defined by integration over boundary of closed curves against a singular kernel function, e.g. single layer potentials. Existence of quadrature formulas drastically simplifies the situation for such operators.

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    $\begingroup$ It might help MOers who want to think about this problem to know that there is significant literature about quadrature domains for arc-length, which I had never heard of before today. I'd recommend that anyone who wants to think about this problem do some reading to find out what is known. (E.g., start by doing a web search for papers on quadrature domains, which often mention literature on quadrature domains for arc-length.) $\endgroup$ – Robert Bryant Mar 20 '15 at 1:34

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