Establish the odd-polygon equilibrium measure formula

Establish that for every odd integer \(N\ge3\) and every \(r>2\), an equilibrium measure for the vertex set \(P_N\) of a regular \(N\)-gon inscribed in the unit circle is supported, up to rotation, on one vertex and the two most distant vertices from it, with the masses and energy given by the formulas stated in Conjecture \(N\) odd.

Background

For $r>2$, Björck’s support theorem reduces equilibrium measures for planar compact sets to supports of at most three extreme points. The authors conjecture that, for an odd regular polygon, the maximizing triple is the asymmetric-looking triple consisting of one vertex and the two vertices farthest from it, with explicitly specified masses.

Exhaustive comparisons of vertex triples on plotting grids support the formula for odd $N\le50$, and the symmetric special case is proved. However, the general reduction from an arbitrary triangle of vertices to the conjectured isosceles configuration is not proved, and finite numerical checks do not establish uniqueness or the conjecture itself.