# Toward a relative q-entropy

## TL;DR

This paper explores the formulation of relative q-entropy for continuous models, proposing the LYZ functional as a promising candidate with notable extremal properties and potential physical implications.

## Contribution

It introduces the LYZ functional as a new approach to defining relative q-entropy for continuous variables, analyzing its extremal distributions and connections to other information measures.

## Key findings

- LYZ functional has extremal properties suitable for relative q-entropy
- Extremizing distributions relate to escort distributions
- Potential physical implications of the LYZ functional are discussed

## Abstract

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.