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Lipschitz continuity, Aleksandrov theorem and characterizations for $$H$$-convex functions. (English) Zbl 1115.49004
The author proves that in stratified groups $$H$$-convex functions locally bounded from above are locally Lipschitz; moreover, he proves that the class of $$v$$-convex functions corresponds to the class of $$H$$-convex functions that are upper semicontinuous, and then they are also Lipschitz. In the class of step 2 groups a more precise result is given; the characterization of locally Lipschitz $$H$$-convex functions as measures whose distributional horizontal Hessian is positive semidefinite.
These results are the generalization of the Euclidean results given by R. M. Dudley [Math. Scand. 41, no. 1, 159–174 (1977; Zbl 0386.46037)] and Yu. G. Reshetnyak [Mat. Sb. (N.S.) 75 (117), 323–334 (1968; Zbl 0176.12001)].

##### MSC:
 49J27 Existence theories for problems in abstract spaces 26A51 Convexity of real functions in one variable, generalizations 26B25 Convexity of real functions of several variables, generalizations 52A41 Convex functions and convex programs in convex geometry
##### Keywords:
convex functions; stratified Lie groups
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##### References:
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