In 1999, Dar conjectured there is a stronger version of the celebrated Brunn-Minkowski inequality. There is a natural relationship between Dar's conjecture and Figalli, Maggi and Pratelli's stability estimate on the isoperimetric inequality. 

We give the solution of Dar's conjecture in dimension 2. The solution inspired us to obtain a non-symmetric log-Brunn-Minkowski inequality. This gives an answer to a question posed by Boroczky, Lutwak, Yang and Zhang.

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