We conjecture that the Parisi functional in the SK model is convex in the functional order parameter and prove a partial result that shows the convexity along one-sided directions. A consequence of this result is the log-convexity of $L_m$ norm for a class or random variables.
"A Question about the Parisi Functional." Electron. Commun. Probab. 10 155 - 166, 2005. https://doi.org/10.1214/ECP.v10-1145