Methods and Applications of Analysis

Volume 28 (2021)

Number 1

Special Issue for the 60th Birthday of John Urbas: Part II

Guest editors: Neil Trudinger and Xu-Jia Wang (Australian National University)

Weak formulation of the MTW condition and convexity properties of potentials

Pages: 53 – 60



Grégoire Loeper (School of Mathematics, Monash University, Clayton, VIC, Australia; BNP Paribas Global Markets, Paris, France; and Ecole Polytechnique, Palaiseau, France)

Neil S. Trudinger (Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia)


We simplify the geometric interpretation of the weak Ma–Trudinger–Wang condition for regularity in optimal transportation and provide a largely geometric proof of the global $c$-convexity of locally $c$-convex potentials when the cost function $c$ is only assumed twice differentiable.


cost function, convexity

2010 Mathematics Subject Classification

35J60, 49Nxx

The authors’ research was partially supported by Australian Research Council Grants DP170100929, DP180100431.

Received 3 September 2020

Accepted 18 November 2020

Published 1 December 2021