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.


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
Full Text: EuDML EMIS