Conservative adaptation consists in a minimal change on a source case to be consistent with the target case, given the domain knowledge. It has been formalised in a previous work thanks to the AGM theory of belief revision applied to propositional logic. However, this formalism is rarely used in case-based reasoning systems. In this paper, conservative adaptation is extended to a more general representation framework, that includes also attribute-value formalisms. In this framework, a case is a class of case instances, which are elements of a metric space. Conservative adaptation is formalised in this framework and is extended to α-conservative adaptation, that relaxes the conservativeness. These approaches to adaptation in a metric space transform adaptation problems to well-formulated optimization problems. A running example in the cooking domain is used to illustrate the notions that are introduced.