In synthetic Lorentzian spaces, the timelike curvature dimension condition TCD_q(K,N) is equivalent to the timelike Brunn-Minkowski inequality TBM_q(K,N) in the q-essentially non-branching case, with a similar equivalence for the entropic version.
Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$-spaces
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The equivalence between timelike Ricci curvature and the timelike Brunn Minkowski inequality on synthetic Lorentzian spaces
In synthetic Lorentzian spaces, the timelike curvature dimension condition TCD_q(K,N) is equivalent to the timelike Brunn-Minkowski inequality TBM_q(K,N) in the q-essentially non-branching case, with a similar equivalence for the entropic version.