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Generalizing the barycenter zero-mean identity to Finsler manifolds

Ascertain whether the identity ∫_X b_γ dμ = 0 for measures μ with a barycenter x0 satisfying b_γ(x0)=0 (where b_γ is the Busemann function of a straight line γ) extends to Finsler manifolds, given that the available splitting is not isometric and the exact product-distance formula is unavailable.

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Background

In the RCD setting, the isometric splitting provides a product structure and a distance formula that yields the identity ∫X bγ dμ = 0 (Lemma 2.4) when x0 is a barycenter and b_γ(x0)=0.

For Finsler manifolds, only a weak (non-isometric) splitting is known; without an exact product-distance formula, the authors are uncertain whether the barycenter zero-mean identity can be generalized.

References

Since this splitting is not isometric, we do not have an exact formula as in eq:prod (consider the case of normed spaces), and it is unclear if Lemma~\ref{lm:0mean} can be generalized.

eq:prod:

dX2(x1,x2)=(bγ(x1)−bγ(x2))2+dY2(ΠY(x1),ΠY(x2))d_X^2(x_1,x_2) =\bigl( b_{\gamma}(x_1)-b_{\gamma}(x_2) \bigr)^2 +d_Y^2\bigl( \Pi_Y(x_1),\Pi_Y(x_2) \bigr)

— A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces (2408.15030 - Brunel et al., 27 Aug 2024) in Section 7 (Further problems), item (C)