A canonical definition of shape

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On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions (2017, methodology): “, page 2467 in Ilmonen and Paindaveine (2011) for a discussion.”
Complex Elliptically Symmetric Distributions: Survey, New Results and Applications (2012, background): “One common normalization [76] is to set .”
OPTIMAL RANK-BASED TESTING FOR PRINCIPAL COMPONENTS (2010, methodology): “…under the whole class of elliptical distributions with finite fourth-order moments can be obtained via a correction factor involving estimated kurtosis coefficients [see Shapiro and Browne (1987) for a general result on the “easy” cases, and Hallin and Paindaveine (2008b) for the “harder” ones].”
Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation (2010, background): “(14) WC functionals are known in the literature as ‘scatter functionals’ (e.g. Dümbgen and Tyler 2005) and ‘shape functionals’ (e.g. Paindaveine 2008; TCDO 2009), among other labels.”