Sufficient conditions are given for the strict convexity of the limit shape in standard first-passage percolation. These conditions involve (1) asymptotic ``straightness'' of the geodesics, and (2) existence of mean-zero limit distributions for the first-passage times.
"Strict Convexity of the Limit Shape in First-Passage Percolation." Electron. Commun. Probab. 8 135 - 141, 2003. https://doi.org/10.1214/ECP.v8-1089