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New properties of convex functions in the Heisenberg group. (English) Zbl 1102.35033
The authors consider the notion of weakly $$H$$-convex function on $$H^n$$ (where $$H^n$$ denotes the Heisenberg group) and prove new properties of weakly $$H$$-convex functions. In particular they investigate the relation between weakly $$H$$-convexity and the generalized subelliptic Monge-Ampère operators in the case $$n=1,2$$ generalizing results, which have been obtained independently by Gutierrez–Montanari [C. E. Gutierrez and A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, preprint, first posted on May 31, 2003, to www.dm.unibo.it/ montanar], and obtaining a theorem of Busemann-Feller-Alexandrov type in the Heisenberg group $$H^n$$, $$n=1,2$$.

##### MSC:
 35H20 Subelliptic equations 26B25 Convexity of real functions of several variables, generalizations 20F18 Nilpotent groups 52A41 Convex functions and convex programs in convex geometry
##### Keywords:
Subelliptic problems; Convexity; Nilpotent groups
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##### References:
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