n has to be >= 1.
OTOH, d has to be >= 0
Why don't we just say "polyhedra" instead of saying "convex polyhedra"? Isn't every polyhedron convex by definition(an intersection of affine
Well, it's really a terminology issue. Some people don't consider polyhedra convex. So for example this person's site: http://www.korthalsaltes.com/ has lots of non-convex polyhedra (eg: great icosahedron).
But yeah, in discrete geometry the definition necessitates convexity.
Most of the eigenvalue optimization theory has been developed for real, symmetric matrices. It is known that such matrices have real eigenvalues. Unsymmetric matrices, on the other hand, have complex eigenvalues in general. It is possible, however, to translate the constraint on the real part of the eigenvalues of a real unsymmetric matrix (say A) to be negative, into a positive definiteness condition on a real symmetric matrix (P) through Lyapunov’s matrix equality (2). Since it is a sufficient and necessary condition for a real symmetric matrix to be positive definite, its eigenvalues to be positive, the condition on the eigenvalues of the "difficult" unsymmetric matrix A is translated into another condition on the eigenvalues of the "not-so-difficult" symmetric matrix P. In order to avoid the potential non-smoothness arising in eigenvalue optimization, interior-point / logarithmic-barrier-transformation techniques have been successfully applied (Ringertz, 1997). For a comprehensive reference of interior-point optimization, see Fiacco and McCormick (1990). Making use of logarithmic and matrix determinant properties, it will be shown that the potentially non-smooth constraints on the eigenvalues of matrix P may be expressed in terms of the determinant of matrix P, which is a smooth function of the optimization variables.
The non-duality of the Brahman,
The non-reality of the world
and the non-difference of the Atman from the Brahman
These constitute the teachings of Advaita