Abstract: I will present a sufficient condition, expressed in terms of the condition numbers of underlying matrices, for a product of positive definite quadratic forms to be convex. The condition is weaker than previously known sufficient conditions, and is also necessary in the case of a product of two forms.

I introduce an equivalence of norms on Banach spaces that is finer than the usual one. In addition to generating the same topology, angularly equivalent norms share certain geometric properties. Equivalent Hilbert space norms are always angularly equivalent, with the constants of equivalence related to the condition number of the matrix relating their inner products