Transactions of the American Mathematical Society
We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli-Garofalo-Nhieu or Lu-Manfredi-Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph.
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First published in Transactions of the American Mathematical Society in 362(8), 2010, published by the American Mathematical Society.
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Capogna, L., Pauls, S. D., & Tyson, J. T. (2010). Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups. Transactions of the American Mathematical Society, 362(8), 4045-4062. https://doi.org/10.1090/S0002-9947-10-04768-9