Centering by translation for free Gibbs measures

Ascertain whether for every continuous potential f: R → R satisfying the growth condition (14) there exists a shift λ ∈ R such that the free Gibbs measure associated with f + λ·id has barycenter zero.

Background

In pursuing the duality between SSFTI and the free functional Santaló inequality, the authors reduce the problem to ensuring the existence of a translation that centers the free Gibbs measure associated with a given potential. They analyze finite-N partition functions and show the existence of centering parameters at the matrix level, but controlling the limit poses difficulties.

This centering conjecture, if resolved, would enable the use of the free functional Santaló inequality under the necessary barycenter condition and could help establish the converse duality to SSFTI.