# Some Convexity Criteria for Differentiable Functions on the 2-Wasserstein Space

arXiv: https://arxiv.org/abs/2306.09120 · HTML: https://mlsanity.com/paper/2306.09120/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2023.

## Built on by (most cited first, 3 of 3)

- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Convex Regularization and Convergence of Policy Gradient Flows under Safety Constraints](https://mlsanity.com/paper/2411.19193/) (2024)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

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.