  In this paper we consider a wide class of generalized Lipschitz extension
problems and the corresponding problem of finding absolutely minimal Lipschitz
extensions. We prove that if a minimal Lipschitz extension exists, then under
certain other mild conditions, a quasi absolutely minimal Lipschitz extension
must exist as well. Here we use the qualifier "quasi" to indicate that the
extending function in question nearly satisfies the conditions of being an
absolutely minimal Lipschitz extension, up to several factors that can be made
arbitrarily small.
