%X We study the convex geometry of certain invariant manifolds, known as carrying simplices, for 3-species competitive Kolmogorov-type maps. We show that if all planes whose normal bundles are contained in a fixed closed and solid convex cone are rendered convex (concave) surfaces by the map, then, if there is a carrying simplex, it is a convex (concave) surface. We apply our results to the May-Leonard map. %I American Institute of Mathematical Sciences %L discovery10049790 %J Discrete and Continuous Dynamical Systems - Series B %A S Baigent %O This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. %V 24 %P 1697-1723 %D 2019 %K Carrying simplex, convexity, concavity, May-Leonard map, invariant manifold %T Convex geometry of the carrying simplex for the May-Leonard map %N 4