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 ??? * * * And in more generality, ask the same question for the supremum over all N non-overlapping, congruent, ellipsoids inside a d-dimensional disk D (bounded by a (d-1)-dimensional sphere): ----- If E_d(N) is the supremum, over all such arrangements, of *the fraction of D's area lying inside any of the N ellipsoids* then: Question_d: ----------- Is E_d(N) a strictly increasing function of N ??? ----- If this is obvious, I'm not sure why yet. —Dan