According to Capogna, Danielli, Tyson & Pauls (An Introduction to the Heisenberg group and the sub Riemannian isoperimetric problem), the isoperimetric best constant in the Heisenberg group is conjectured (by P. Pansu) to be 3^{3/4}/4\sqrt{\pi} (attained for the bubble set). However, from an unpublished paper of Capogna (Isoperimetric inequalities in the Heisenberg group and in the plane), it follows from the formulae therein that the best constant is 3^{3/4}/4^{5/4}\sqrt{\pi}. Which is the correct one?
Dear Michael and Rolf, I just wrote to Capogna. If he answers, I will publish his answer asap.
Hello, you could probably find the answer in the following paper by G.P. Leonardi and S. Rigot :
http://cvgmt.sns.it/media/doc/paper/1372/leorig01.ps
I got an answer from Roberto Monti in Padova. Both values are correct, the difference is due to the different normalaisation of the coefficients of the invariant vector fields of the Heisenberg group.
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