When is itself a compact set, such as the unit hypercube for , we can make the uniform distribution. Then, so long as the manifold is compact for all 3.1, then will be a proper uniform distribution for all , which has maximum entropy. Alternatively, when is infinite in extent, the manifold can be forced to be compact by the inclusion of an energy statistic in (first proposed in ). The solution for compact and the solution for unbounded are formalized by the following two theorems.
The second case considers when is not compact.
The highest-entropy distribution, which is found by maximizing the entropy of over , is denoted by . Intuitively, has the MaxEnt property because all samples generated by for a given are equally likely (the uniform distribution, which has maximum entropy on a compact set). This is a result of the fact that for a given , all samples on the manifold have constant value on , making also constant on the manifold, implying that the manifold distribution is in fact the uniform distribution. The reader is referred to a previous article for additional details of the proof .