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I've been digging deep for methods to test multivariate normality (MVN) of data of about 10-12 dimensions (usual methods like the Shapiro-Wilk don't seem to have enough power for this). I recently came upon this publication, which in Remark 3 on the final page suggests that for a datapoint $X_k$ of MVN data $\{X_1,X_2,...,X_n\}$ the projections of all other $X_i$ datapoints on $X_k$ (given by $X_k^TX_i$ for $X_i$) are guaranteed to be normally distributed. Do I interpret this correctly? If I do can I use this to develop a test for multivariate normality based on this argument?

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