Oleksandr Metelichenko (University College London & Odessa University)
Isodiametric problem in the groups with dilations
Abstract. We show that the isodiametric inequality (a metric ball maximizes the volume for the given diameter) fails in the Heisenberg group equipped with the Carnot-Carath\'eodory (geodesic) metric and give estimates for the ratio of the volume of the ball and the maximal volume of a set of the same diameter. We also give some examples of the similar nature in the additive group $\R^n$ with non-isotropic dilations.


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