Long-wavelength geoid anomalies due to lateral variations in the density or thickness of a thin layer in local isostatic equilibrium on the surface of a sphere depend sensitivity on the assumed state of stress within the layer. A number of common intuitive definitions of local isostasy generally believed to be essentially identical are associated with quite different states of isostatic stress, and the corresponding theoretical geoid anomalies can vary by more than a factor of 2. This sensitivity of the theoretical anomaly to the exact definition of local isostasy constitutes an obstacle to any proposed program of interpreting or exploiting observed global isostatic geoid anomalies such as those associated with the aging of the oceanic lithosphere.