Affine transformations of circle and sphere
  A non-degenerate two-dimensional linear operator $\varphi$ transforms the unit circle into ellipse. Let $p$ be the length of its bigger axis and $q$ -- the length of smaller. We can define the deformation coefficient $k(\varphi)$ as $q/p$. Analogously, if $\varphi$ is a non-degenerate three-dimensional operator, then it transforms the unit sphere into ellipsoid. If $p>q>r$ are lengths of its axes, then deformation coefficient $k(\varphi)$ will be defined as $r/p$. In this work we compute the mean value of deformation coefficient in two-dimensional case and give an estimation of mean value in three-dimensional case. 
Irina Busjatskaja and Yury Kochetkov
math.MG