Possibility theory provides a formal system for support representation and combination appropriate for diagnosis, classification and decision analysis. This paper examines the semantics, the measurement, and the combination of possibilistic support for evidential reasoning. It is shown that normalization is incompatible with the combination of possibilistic support. Using the ‘fuzzy sets as points’ representation, possibility distributions may be represented as points in an n-dimensional cube. The metric properties of this space are used to define the fundamental concepts of fuzzy set theory. Employing this representation, geometric measures of uncertainty in possibilistic support are proposed. The nonspecificity of a possibility distribution, the degree to which the support assignment fails to designate a single hypothesis, is defined as the minimal distance to a distribution that uniquely designates a single hypothesis. The conflict in the support is determined by the degree of subnormality of a distribution. The relationships between geometric nonspecificity and conflict are detailed and compared with the properties of U-uncertainty, the standard information based measure of possibilistic uncertainty.