Industrial processes typically involve heating and cooling fluids via networks of heat exchangers which reuse excess process heat onsite. Optimally synthesising these networks of heat exchangers is a mixed-integer nonlinear optimisation problem with nonlinear terms including bilinear stream mixing, concave cost functions, and the logarithmic mean temperature difference (LMTD), which characterises the nonlinear nature of heat exchange. LMTD is typically associated with numerical difficulties, but, after adding the limits, this manuscript proves the strict convexity of LMTDβ, β < 0, and also characterises the shape of the function for all β ≤ 1. These proofs motivate why previous, heuristic-based approaches work best when the problem is reformulated to move the LMTD terms into the objective. The convexity results also lead to an effective algorithm bounding the simultaneous synthesis model SYNHEAT from the online test set MINLPLib2; this algorithm solves a series of mixed-integer linear optimisation problems converging to the global objective value of the original problem.