  The distance-decay function of the geographical gravity model is originally
an inverse power law, which suggests a scaling process in spatial interaction.
However, the distance exponent of the model cannot be explained with the ideas
from Euclidean geometry. This results in what is called dimension dilemma. In
particular, the gravity model based on power law could not be derived from
general principles by traditional ways. Consequently, a negative exponential
function substituted for the inverse power function to serve for a
distance-decay function for the gravity model. However, the exponential-based
gravity model goes against the first law of geography. This paper is devoted to
solve these kinds of problems by mathematical reasoning and empirical analysis.
First, it can be proved that the distance exponent of the gravity model is
essentially a fractal dimension. Thus the dimensional dilemma of the
power-based gravity model can be resolved using the concepts from fractal
geometry. Second, the exponential function indicates locality and localization,
which violates the basic principle of spatial interaction. The power function
implies action at a distance, which is the necessary condition of geographical
gravitation. Third, the gravity model based on power law decay can be derived
from the entropy- maximizing principle by introducing a proper postulate. The
observational data of China's cities and regions are employed to verify the
theoretical inferences, and the results support power-law distance decay. A
conclusion can be reached that the preferred form of geographical gravity model
is its original form, which is based on an inverse power law rather than a
negative exponential law.
