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The theorem of Busemann-Feller-Alexandrov in Carnot groups. (English) Zbl 1071.22004
The authors prove a version of the Busemann-Feller-Alexandrov theorem for the class of weakly $$H$$-convex functions in Carnot groups. Precisely, let $$\mathbf G$$ be a Carnot group of step $$r=2$$ with a system $$X_1,\dots,X_m$$ of bracket generating left-invariant vector fields. If $$u\in C({\mathbf G})$$ is a weakly $$H$$-convex function, then the horizontal second derivatives $$X_iX_ju$$ exist at a.e. point in $$\mathbf G.$$ More precisely, for $$dg$$-a.e. point $$g_0\in{\mathbf G}$$ there exists a polynomial of weighted degree $$\leq 2,$$ $$P_u(g;g_0),$$ such that $\lim_{g\to g_0}\frac{u(g)-P_u(g;g_0)}{d(g,g_0)^2}=0.$

##### MSC:
 2.2e+21 General properties and structure of other Lie groups 2.2e+31 Analysis on real and complex Lie groups
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