The centroid, for triangle side length of 1, is ~0.419 (according to Mathematica), but ~0.544 (according to Antiprism). Not sure what I've done differently,


I see the difference now, ~0.419 is the centroid of volume and ~0.544 is the centroid of the vertices.

It is not too difficult to ...

The centroid, for triangle side length of 1, is ~0.419 (according to Mathematica), but ~0.544 (according to Antiprism). Not sure what I've done differently, or am missing.
The bottom three vertices are at height 0.0, the second three are at height 0.86..., the centroid will be higher than 0.43 ...

Going back to the canonical construction of the solid and its dual, we can readily see that your Chestahedron/ellipsoid construction is a simple stretching of the canonical construction and that symmetries relating to the stretching axis are preserved. In particular, the relative volumes and ...

The term "dual" isn't well defined for polyhedra.

If the base model has central symmetry and you reciprocate in a sphere with this centre then you will probably get general agreement that the result is the shape of the dual. If the base model also has a midsphere and you reciprocate in this then ...

The canonical form of the chesterhedron is geometrically self-dual by reciprocation in a sphere.

conv_hull Chestahedron-Pts.off | canonical -l 15 | pol_recip -a -c M -n 10000 | antiview

http://antiprism.com/misc/tri_antiherm_can_dual.png

The chesterhedron itself may be geometrically self-dual ...