Hi Dan,
It would help if you distribute an example STL file. Your question may be answered using regions. See for example:
Integrate Regions
You may also need Eigensystem or Eigenvectors. Hope it helps.
Brad
Edit:
Following Christopher's example, we write a few expressions to make some connections to the type of expressions you would see in an undergrad physics textbook.
r = DiscretizeGraphics[ExampleData[{"Geometry3D", "SedanCar"}]];
Vol = Integrate[1, {x, y, z} \[Element] r];
COM = Integrate[#, {x, y, z} \[Element] r]/Vol & /@ {x, y, z};
Moments =
Outer[Integrate[#1 #2, {x, y, z} \[Element] r] &, {x, y, z} -
COM, {x, y, z} - COM];
Chop[Subtract[RegionCentroid[r], COM], 10^(-9)]
MomentOfInertia[r] // MatrixForm
Moments // MatrixForm
The matrices are the same up to a multiple of the identity matrix, so we calculate the same eigenvectors:
Eigenvectors[ Moments ] // MatrixForm
Eigenvectors[ MomentOfInertia[r] ] // MatrixForm
As for choosing the minimum, I think the mathematical definitions involve the corresponding eigenvalues. This math approach is useful to know in a few cases where the solid is nearly spherical or has approximate circular symmetry. In many cases, such as the example of a car, Christopher's common sense method of plotting the solid and looking for the long axis is the usual for physicists.
