I'm still not sure, but it looks as though for N = 3 ellipses in a unit disk if you chop the inscribed hexagon into 3 60º - 120º rhombi of side = 1, then possibly the largest ellipses in such a rhombus add up to more area than the largest congruent circular disks that can all 3 exist disjointly in D. But that *might* be the only case... —Dan James Buddenhagen wrote: ----- I don't seem to have good intuition on this, but do you have an example where N congruent ellipses covers more than N congruent circles? What about very eccentric ellipses whose major axes are like spokes of a wheel? On Mon, May 6, 2019 at 12:44 PM Dan Asimov <dasimov@earthlink.net> wrote:

I probably should have asked if E(N) is monotonic *decreasing*, since E(1) = 1 and for N > 1, E(N) < 1. But after N = 2 it's hard to see what happens.

Also, for N large almost all ellipses will not be near the boundary circle, so it seems likely that as N —> oo the configurations will approach close-packed disks in the plane, so the density will approach π/sqrt(12).

—Dan

----- Let E(N) be the maximum possible fraction of a 2-disk D's area that is occupied by N non-overlapping, congruent ellipses in D.

OK, the supremum.

E(N) seems pretty hard to determine explicitly for each N. But one thing that seems not immediately obvious to me is:

Question: --------- Is E(N) an increasing function of N ??? -----