Planar packings and mappings related to certain minmax problems

Abstract

For any integer N greater than or equal to I, we obtain the extremal values of the minmax problem for exponential sums, f (N) := min(a1 real) max {\Sigma(n=1)(N) e(1an)\,\Sigma(n=1)(N) e(iNan)\} In particular, the two extremal problems f (3) and max(a,b,c is an element of [0,2pi]) min {\(e(ia) - e(ib)) (e(ib) - e(ic)) (e(ic) - e(ia))\ , \ (e(i3a) - e(i3b)) (e(i3b) - e(i3c))(e(i3c) - e(i3a))\} are reduced to the problems about the packing of certain convex sets in the plane. This packing method also can be used to solve some other extremal problems.