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For fully nonlinear k-Hessian operators on bounded strictly (k-1)-convex domains of Euclidian space, a characterization of the principal eigenvalue associated to a k-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of elliptic branches in the sense of Krylov [TAMS’95] that correspond to selecting k-convex functions. Moreover, the associated principal eigenfunction is constructed by an iterative viscosity solution technique, which exploits a compactness property coming from the establishment of a global Hölder continuity property for the approximating equations. This is joint work with Isabeau Birindelli.