In this paper, the authors characterise the maximal convex subsets of the (non-convex) rate region in 802.11 WLANs. In addition to being of intrinsic interest as a fundamental property of 802.11 WLANs, this characterisation can be exploited to allow the wealth of convex optimisation approaches to be applied to 802.11 WLANs. They establish a number of fundamental convexity properties of the rate region of 802.11 WLANs. Firstly, they establish a simple constraint that determines the station transmission attempt probabilities on the rate region boundary. Secondly, they show that, while the rate region is non-convex, its complement in the positive orthant is strictly convex.