SOME PROJECTIVE DISTANCE INEQUALITIES FOR SIMPLICES IN COMPLEX PROJECTIVE SPACE
Mark Edward Fincher, Heather Olney, William Cherry · 2014
Abstract. We prove inequalities relating the absolute value of the determi-nant of n + 1 linearly independent unit vectors in Cn+1 and the projective distances from the vertices to the hyperplanes containing the opposite faces of the simplices in complex projective n-space whose vertices or faces are deter-mined by the given vectors. A basis of unit vectors in Cn+1 determines the vertices (or the faces) of a simplex in n-dimensional complex projective space. For reasons originally motivated by an inequality in complex function theory proven by Eremenko and the third author [CE], we investigated the relationship between the determinant of the vectors form-ing the basis and the projective distances from each vertex of the simplex to the hyperplane containing the face of the opposite side. We show that if dmin denotes the minimum of these projective distances and if D denotes the determinant of the basis vectors, then dnmin ≤ |D | ≤ dmin. Acknowledgments. Surya Raghavendran, during a research experiences for un-dergraduates project supervised by the third author and funded by a SUMS fel-