Sufficient conditions for existence of non-trivial Hausdorff measure on Busemann concave spaces

Determine explicit sufficient conditions under which a Busemann concave metric space admits a non-trivial Hausdorff measure of some integer dimension, i.e., establish criteria guaranteeing that there exists n in N with 0 < H^n(X) < ∞ for the space X equipped with its metric d.

Background

The paper develops a structure theory for finite-dimensional Busemann concave spaces under additional assumptions such as S-concavity and local semi-convexity, proving the existence of a non-trivial integer-dimensional Hausdorff measure in that setting. However, beyond these hypotheses, the general existence of a non-trivial Hausdorff measure on Busemann concave spaces is not settled.

Kell’s earlier work analyzed metric measure properties of Busemann concave spaces conditional on the existence of a non-trivial Hausdorff measure; identifying intrinsic geometric conditions that guarantee such measures would close a foundational gap and broaden the applicability of measure-contraction and rectifiability results.