Summary: We study the transportation problem on the unit sphere \(S^{n-1}\) for symmetric probability measures and the cost function \(c(x,y) = \log\frac{1}{\langle x, y \rangle}\). We calculate the variation of the corresponding Kantorovich functional \(K\) and study a naturally associated metric-measure space on \(S^{n-1}\) endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric log-Minkowski problem and prove that \(K\) satisfies the following analog of the Gaussian transportation inequality for the uniform probability measure \({\sigma}\) on \(S^{n-1}:\frac{1}{n}\operatorname{Ent}(\nu)\geq K({\sigma},\nu)\). It is shown that there exists a remarkable similarity between our results and the theory of the Kähler-Einstein equation on Euclidean space. As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.