We show that a differentiable function on the 2-Wasserstein space is geodesically convex if and only if it is also convex along a larger class of curves which we call ‘acceleration-free’. In particular, the set of acceleration-free curves includes all generalised geodesics. We also show that geodesic convexity can be characterised through first- and second-order inequalities involving the Wasserstein gradient and the Wasserstein Hessian. Subsequently, such inequalities also characterise convexity along acceleration-free curves.

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2-Wasserstein 空间上可微函数的一些凸性准则

我们证明，2-Wasserstein 空间上的可微函数是测地凸函数，当且仅当它也沿着我们称为“无加速”的更大类曲线凸时。特别地，无加速度曲线组包括所有广义测地线。我们还表明，测地凸性可以通过涉及 Wasserstein 梯度和 Wasserstein Hessian 的一阶和二阶不等式来表征。随后，这种不等式也表征了沿无加速度曲线的凸性。