Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse.
It seems like there should be an elegant proof.
Note that if degenerate ellipses, i.e. line segments, are allowed, then the conclusion does not follow -- radial segments can be used that get arbitrarily close to the circle's center in every sector.
(This question was inspired by this one about filling the plane with parabolas.)
This is hard to make rigorous, but if assume by contradiction that all the space inside $C$ is taken, then there are couples and triples of ellipses arbitrarily close. Assume that we have $3$ ellipses very close each other, like in figure:
then they define a region $D$ with a concave boundary with the property that any entering ellipse cannot have a minor axis greater than the separation between the ellipses. Hence I would bet that the measure of the subset of $D$ taken by entering ellipses cannot exceed the sum of the areas of the three depicted trapezoids, and such a sum is stricly less than $\mu(D)$, hence in $D$ there must be a neighbourhood of a point that is not taken by any ellipse, as wanted.
However, this argument just gives an idea and a bet, and it is still far from being a proof.