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Brunn-Minkowski and isoperimetric inequality in the Heisenberg group. (English) Zbl 1075.49017

The author proves that the Brunn-Minkowski inequality is not true in the one-dimensional Heisenberg group. The Brunn-Minkowski inequality in the setting of Heisenberg group reads like \[ | A\cdot B| ^{1/4}\geq | A| ^{1/4}+| B| ^{1/4}, \] with \(A,B\subset R^3\) any bounded open sets, where \(\cdot\) is the group operation and \(4\) is the homogeneous dimension. The author shows that if the Brunn-Minkowski inequality holds, this would imply the isoperimetric property of Carnot-Carathéodory balls, a property that is known to be false.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
28A75 Length, area, volume, other geometric measure theory
43A80 Analysis on other specific Lie groups
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