  Various concepts of mean shape previously unrelated in the literature are
brought into relation. In particular for non-manifolds such as Kendall's 3D
shape space, this paper answers the question, for which means one may apply a
two-sample test. The answer is positive if intrinsic or Ziezold means are used.
The underlying general result of manifold stability of a mean on a shape space,
the quotient due to an isometric action of a compact Lie group on a Riemannian
manifold, blends the Slice Theorem from differential geometry with the
statistics of shape. For 3D Procrustes means, however, a counterexample is
given. To further elucidate on subtleties of means, for spheres and Kendall's
shape spaces, a first order relationship between intrinsic,
residual/Procrustean and extrinsic/Ziezold means is derived stating that for
high concentration the latter approximately divides the (generalized) geodesic
segment between the former two by the ratio $1:3$. This fact, consequences of
coordinate choices for the power of tests and other details, e.g. that
extrinsic Schoenberg means may increase dimension are discussed and illustrated
by simulations and exemplary datasets.
