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Convex functions on Carnot groups. (English) Zbl 1124.49024
Summary: 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.

##### MSC:
 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 35J70 Degenerate elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems 22E30 Analysis on real and complex Lie groups 49N60 Regularity of solutions in optimal control
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