We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in first-passage percolation, as a functional on the space of distribution functions. The present counterexample only works for first-passage percolation on $Z^d$ for $d$ large.
"On the Non-Convexity of the Time Constant in First-Passage Percolation." Electron. Commun. Probab. 1 1 - 6, 1996. https://doi.org/10.1214/ECP.v1-971