An electrostatic skeleton problem for polygons: B. Gustafsson showed that every polygon has an "mother body", a skeleton (measure supported on a compact set with empty interior and connected complement) that generates the same exterior (logarithmic) potential as area measure on the polygon.

E. Saff asked whether the same is true for equilibrium measure of the polygon. If there is such a skeleton he calls it a "Madonna body".

The question is stated with an example worked out in Section 4 of my recent preprint with V. Totik.

Two points of interest: (i) The mother body of a polygon is always a union of line segments, while the Madonna body can have curved pieces. (ii) It is not even known if every convex polygon has a Madonna body.

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