Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ forms a group.
We define $(x,\gamma)=x(\gamma)\sqrt(\gamma')$ where $\gamma'$ is the derivative of $\gamma$ and by varying $\gamma$ over $\Gamma$ we get a class of function.
Suppose we are given with the following two conditions for two distinct member of $x_1$ and $x_2$ of $x$ as \begin{eqnarray} ||\mu-x^*_1||=\inf_{\gamma\in \Gamma}||\mu-(x_1,\gamma)||,\\ \mbox{and}\;\; ||\mu-x^*_2||=\inf_{\gamma\in \Gamma}||\mu-(x_2,\gamma)||, \end{eqnarray} where ||.|| is the usual $L_2$ norm of function.
Then I am trying to prove, whether the following is true $$||\mu-(x^*_1+x^*_2)||=\inf_{\gamma\in \Gamma}||\mu-((x_1+x_2),\gamma)||$$
Any helpful comments or answer are welcome. Thanks