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A new characterization of convexity in free Carnot groups. (English) Zbl 1272.31009
Summary: A characterization of convex functions in $$\mathbb R^N$$ states that an upper semicontinuous function $$u$$ is convex if and only if $$u(Ax)$$ is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix $$A$$. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups $$\mathbb G$$ when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps $$x\mapsto Ax$$ of the Euclidean case must be replaced by suitable group isomorphisms $$x\mapsto T_A(x)$$, whose differential preserves the first layer of the stratification of $$\operatorname{Lie}(\mathbb G)$$.

##### MSC:
 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 26B25 Convexity of real functions of several variables, generalizations 43A80 Analysis on other specific Lie groups 35J70 Degenerate elliptic equations
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