Geodesic concavity of microstates free entropy for full types

Establish whether the microstates free entropy for full types χ_full is concave along Wasserstein geodesics in the type space S_{m}(T_ω), i.e., whether t ↦ χ_full(μ_t) is concave for 0 < t < 1 when (μ_t) is the d_{W,full}-geodesic induced by an optimal coupling between given endpoints.

Background

The authors prove lower bounds for χ_full along Wasserstein geodesics and a Lipschitz continuity modulus, but a full geodesic concavity statement—analogous to classical entropy—is not established.

They explain that current techniques rely on transport maps that are only bi-Lipschitz, without sufficient differentiability to apply change-of-variables formulas involving Jacobians. This lack of smoothness in definable transport functions blocks a direct approach to proving concavity.

Resolving geodesic concavity would significantly advance free information geometry by paralleling classical results where entropy is geodesically concave under optimal transport.