For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right \rangle d(y_i,y_j)$ over $y$ where $d(\cdot)$ is the squared Euclidean distance. The minimization is obtained over orthonormal constraints. i.e, the matrix $Y_{k \times n}$ whose rows are $y_i,y_j,\ldots,y_k$ are orthonormal. Is the result obtained by this mapping isometric to the points $x$?
Can this be proved for isometry or non-isometry? The usage of inner-products and euclidean distances, (which is a metric induced by the inner product norm) makes me feel that this mapping is isometric. Am not sure and am looking for a proof.
You may instead answer this question over a map that preserves $\|\langle x_i,x_j\rangle- d(y_i,y_j)\|$ instead of the above function $f(\cdot)$ if you find this easier to work around inorder to prove/check for isometry. In a broader sense, am looking for a proof for isometry for a map that preserves the inner-products as euclidean-distances.