Title: Some Convexity Criteria for Differentiable Functions on the 2-Wasserstein Space Show Chinese title

Title: 关于可微函数在2-瓦瑟斯坦空间的一些凸性准则

Abstract: 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. Show Chinese abstract

Abstract: 我们证明了定义在2-Wasserstein空间上的可微函数当且仅当沿一类我们称为“无加速”的更广曲线族上是凸的时，在测地线上也是凸的。特别地，无加速曲线族包含了所有广义测地线。此外，我们还表明测地线凸性可以通过涉及Wasserstein梯度和Wasserstein Hessian的一阶和二阶不等式来刻画。进一步地，这些不等式也刻画了沿无加速曲线的凸性。