What is the meaning of the quantity:
Is there, for example, a geometric explanation? Is there a term for it in statistics?
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It is the square root of the determinant of the covariance matrix (between $X$ and $P$). The determinant of the covariance matrix is called as Generalized Variance, which quantifies the co-variability of multivariate random variables to a scalar. What you write is the square root of it, so I believe it won't be too odd to call it as Generalized Deviation.