Some convexity criteria for differentiable functions on the 2‐Wasserstein space

Parker, Guy

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.

Citation

Parker, G. (2024). Some convexity criteria for differentiable functions on the 2‐Wasserstein space. Bulletin of the London Mathematical Society, 56(5), 1839–1858. https://doi.org/10.1112/blms.13030

• Journal Article Type: Article
• Acceptance Date: Feb 4, 2024
• Online Publication Date: Mar 17, 2024
• Publication Date: Mar 17, 2024
• Deposit Date: May 22, 2024
• Publicly Available Date: May 22, 2024
• Journal: Bulletin of the London Mathematical Society
• Print ISSN: 0024-6093
• Electronic ISSN: 1469-2120
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 5
• Pages: 1839–1858
• DOI: https://doi.org/10.1112/blms.13030
• Public URL: https://durham-repository.worktribe.com/output/2346177