On a remark by Ohsawa related to the Berndtsson–Lempert method for $L^2$-holomorphic extension

. In [15, Remark 4.1], Ohsawa asked whether it is possible to prove Theorem 4.1 and Theorem 0.1 in [15] using the Berndtsson-Lempert method. We shall answer Ohsawa’s question affirmatively in this paper. Our approach also suggests to introduce the Legendre-Fenchel theory and weak psh-geodesics into the Berndtsson-Lempert method.