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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
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