Two remarks on the Suita conjecture

It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1+\varepsilon_-}$-smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain.Comment: Ann. Pol. Math. (to appear

Kobayashi-Royden pseudometric vs. Lempert function

We give an example showing that the Kobayashi-Royden pseudometric for a non-taut domain is, in general, not the derivative of the Lempert function.Comment: An essential mistake has been corrected in v2. Improved example in v3. v4: minor changes according to the referee remarks (Ann. Mat. Pura Appl.