This paper applies a framework which addresses the vagueness of poverty. The 'core poor' are those who are unambiguously poor. In applying the framework we use Sen's capability approach and results from a recent survey. These results suggest that some South Africans set tough standards for someone to qualify as poor. Even by these standards, our lower bound estimate of core poverty is higher than existing estimates of the 'most deprived' and 'ultrapoor'. This result is sensitive to the criteria used in applying the framework, though other results are more robust. While there is evidence that respondents adapted to their living conditions, it was not merely those who were deprived in specific dimensions who endorsed very low cut-offs in those dimensions.