What would then be the way to determine a true dimensionality of a configuration of points $X\in\mathbb{R}^{n\times k}$ based on its Gram matrix $G=XX^T$? Different shifts would imply different ranks of $G$ that correspond to different dimensionalities. Consider, for instance, the paper http://convexoptimization.com/TOOLS/Gower1.pdf (Sect. 3, very short, please consider)

If I understand correctly, as long as the shifts are of the form $X'=PX$, $P=I_n-1_nw^T$, where $1_n^Tw=1$, the rank of $G'=X'(X')^T$ is the true dimensionality of $X$? If you consider the 2x2 example from the above paper, a shift where one point is the origin reduces the dimensionality to 1D, which is the true dimensionality (same holds for a centroid origin) Does that mean that, in order to recover the "true" configuration, one is restricted to the above form of the shift?

What would then be the way to determine a true dimensionality of a configuration of points $X\in\mathbb{R}^{n\times k}$ based on its Gram matrix $G=XX^T$? The "true" dimensionality refers to the minimum number of dimensions needed to represent a configuration (eg, a line is 1D, but can be represented in 2D).

Different shifts would imply different ranks of $G$ that correspond to different dimensionalities. Consider, for instance, the paper http://convexoptimization.com/TOOLS/Gower1.pdf (Sect. 3, very short, please consider)

If I understand correctly, as long as the shifts are of the form $X'=PX$, $P=I_n-1_nw^T$, where $1_n^Tw=1$, the rank of $G'=X'(X')^T$ is the true dimensionality of $X$? If you consider the 2x2 example from the above paper, a shift where one point is the origin reduces the dimensionality to 1D, which is the true dimensionality (same holds for a centroid origin) Does that mean that, in order to recover the "true" configuration, one is restricted to the above form of the shift?