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Notion of convexity in Carnot groups. (English) Zbl 1077.22007
The aim of this interesting paper is to study appropriate notions of convexity in the setting of Carnot groups $$G$$. First, the notion of strong $$H$$-convexity is examined. Some arguments showing that the concept is to restrictive are presented. Then the notion of weakly $$H$$-convex functions is defined. A function $$u:G\rightarrow {\mathbb R}$$ is weakly $$H$$-convex if for any $$g\in G$$ and every $$\lambda\in [0,1]$$ we have $u(g\delta_\lambda(g^{-1}g'))\leq (1-\lambda_u(g) + \lambda u(g')),$ where $$\delta_\lambda$$ is a group dilation and $$g'$$ is an element of the horizontal plane $$H_g$$ passing through $$g$$. It is proved that a twice differentiable function is weakly $$H$$-convex iff its symmetrized horizontal Hessian is positive semi-definite at any $$g\in G$$. This is the subelliptic counterpart of the classical characterization of convex functions. The intrinsic gauge in any group of Heisenberg type is weakly $$H$$-convex. Moreover, a weakly $$H$$-convex function is Lipschitz continuous with respect to the sub-Riemannian metric of $$G$$. The main result of the paper says that the supremum of the absolute value of a weakly $$H$$-convex continuous function over any ball can be estimated from above by the mean value of the absolute value. The local boundedness, the continuity on effective domains of weakly $$H$$-convex functions as well as their relations to fully nonlinear differential operators in the sub-Riemannian setting are studied.

##### MSC:
 22E25 Nilpotent and solvable Lie groups 35A30 Geometric theory, characteristics, transformations in context of PDEs
##### Keywords:
Carnot groups; convex functions; fully nonlinear PDE
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