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

Abstract

AbstractWe 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.