Triangles generated by powers of triplets on unit circle

Abstract

Le t a, 13, y be th ree d is tin c t complex numbers of mod ulus 1. It is shown that th ere is essent ially one e xception to the followin g s tat e ment: For some posi t ive int eger Ill , 0 is in th e c losed convex hull of a"', 13"' , y"'. The exception occurs for the normalized tripl e 1, e'l 1T i/7, e2krri/7, whe re k = 3 or 5. Thi s qu es tion was motivated by the proble'm of determining when a pos itive intege r III and a nonze ro n X l vee'tor X e xist such tha i X * A IIIX = 0, whe re A is a given matrix of M,,(C). Key word s: Convex hull; lInit ci rcle; \V ey l's Theorem.