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This investigation shows a link between nonnegative sectional curvature and matrix displacement convexity, suggesting implications for variational inequalities.
We show that the sectional curvature of a Riemannian manifold is nonnegative if, and only if, the entropy functional is matrix displacement convex. As an application we obtain intrinsic dimensional evolution variational inequalities, and the corresponding Wasserstein contraction along heat flows, which improve on their dimensional counterparts.
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Aishwarya et al. (2025) studied this question.
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