Metelichenko, O.B.; (2006) The isodiametric inequality in locally compact groups. Doctoral thesis , University of London.

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Abstract

In this thesis it is shown that the isodiametric inequality fails for Carnot-Caratheodory balls in the Heisenberg group W (n E N). Estimates for the ratio of the volume of a ball to the maximal volume of a set of the same diameter are established in this group, and the set of the maximal volume is also found among all sets of revolution about the vertical axis having the same diameter. Results of the similar nature are obtained in the additive group Rn+1 (n G N) with non-isotropic dilations. Using a connection between the isodiametric problem and the Besicovitch 1/2-problem it is proved that the generalized Besicovitch 1/2-conjecture fails in the Heisenberg group HP (1 < n < 8) of the Hausdorff dimension 2n+2 and the additive group Rn+1 (n £ N) having non-isotropic dilations and integer Hausdorff dimension greater than or equal to n + 2. But the 1-dimensional case is shown to be exceptional - the generalized Besicovitch 1/2-conjecture is true in any locally compact group which is equipped with an invariant metric, its Haar measure and has the Hausdorff dimension 1. A question about the relation among the Hausdorff, the spherical and the centred Hausdorff measures of codimension one restricted to a smooth surface is also investigated in the Heisenberg group M1. It is proved that these measures differ but coincide up to positive constant multiples, estimates for which are found.

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