# A characterisation of convex order using the 2-Wasserstein distance

@inproceedings{Wiesel2022ACO, title={A characterisation of convex order using the 2-Wasserstein distance}, author={Johannes Wiesel and Eric Zhang}, year={2022} }

. We give a new characterisation of convex order using the 2-Wasserstein distance W 2 : we show that two probability measures µ and ν on R d with ﬁnite second moments are in convex order (i.e. µ (cid:22) c ν ) iﬀ holds for all probability measures ρ on R d with bounded support. Our proof of this result relies on a quantitative bound for the inﬁmum of (cid:82) f dν − (cid:82) f dµ over all 1-Lipschitz functions f , which is obtained through optimal transport duality and Brenier’s theorem. We use…

## 2 Citations

### Martingale Transports and Monge Maps

- Mathematics
- 2022

It is well known that martingale transport plans between marginals µ (cid:54) = ν are never given by Monge maps—with the understanding that the map is over the ﬁrst marginal µ , or forward in time.…

### A short proof of the characterisation of convex order using the 2-Wasserstein distance

- Mathematics
- 2022

. We provide a short proof of the intriguing characterisation of the convex order given by Wiesel and Zhang [5].

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