We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.
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Petri Juutinen, Guozhen Lu, Juan J. Manfredi, Bianca Stroffolini, Convex functions on Carnot groups. Rev. Mat. Iberoam. 23 (2007), no. 1, pp. 191–200