We address the question and related controversy of the formulation of the $q$-entropy, and its relative entropy counterpart, for models described by continuous (non-discrete) sets of variables. We notice that an $L_p$ normalized functional proposed by Lutwak-Yang-Zhang (LYZ), which is essentially a variation of a properly normalized relative R\'{ e}nyi entropy up to a logarithm, has extremal properties that make it an attractive candidate which can be used to construct such a relative $q$-entropy. We comment on the extremizing probability distributions of this LYZ functional, its relation to the escort distributions, a generalized Fisher information and the corresponding Cram\'{ e}r-Rao inequality. We point out potential physical implications of the LYZ entropic functional and of its extremal distributions.
Categories: cond-mat.stat-mech gr-qc hep-th math-ph math.MP