A well-known result by Lindenstrauss is that any two-dimensional normed space can be isometrically imbedded into $L_1(0,1)$. We provide an explicit form of a such an imbedding. The proof is elementary and self-contained. Applications are given concerning the following: (i) explicit representations of the moments of the norm of a random vector $X$ in terms of the characteristic function and the Fourier--Laplace transform of the distribution of $X$; (ii) an explicit and partially improved form of the exact version of the Littlewood--Khinchin--Kahane inequality obtained by Latała and Oleszkiewicz; (iii) an extension of an inequality by Buja--Logan--Reeds--Shepp, arising from a statistical problem.