  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 $S^{n-1}$:
  $\frac{1}{n} Ent(\nu) \ge K({\sigma}, \nu)$. It is shown that there exists a
remarkable similarity between our results and the theory of the
K{\"a}hler-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.
