Vtyjkyuk

Given two points $y,z$ on the circle, find $x$ on the shortest arc between $y$ and $z$ such that $|x-y|^2+|x-z|^2$ is minimized. The Langrange conditions give $-y-z+lambda x=0$ which yields $x=fracy+z$. Note that the minimizer satisfies $|x-y|=|x-z|$.

I want to generalize this result to a manifold embedded in a high-dimensional sphere, $Msubset S^dsubset R^d+1$ for some large $d$ (e.g. the Stiefel manifold or the Grassmannian). Given $y,zin M$ (with $|y|=|z|=k$ in the Frobenius norm), let $T==$. Under what conditions on $M$ can we conclude that there is a local solution to $min_xin M|x-y|^2+|x-z|^2$ which belongs to $T$? It is true when $M$ is a circle (see above). When $M$ is $S^n$ it is true for $x=fracy+z$, but not the other points of $T$.

Is it true when $M$ is the Stiefel manifold $V_p,n=Xin R^ntimes p$? Using the Lagrange conditions I get $-Y-Z+XLambda=0$. Then $Lambda=-X^T(Y+Z)$. Assuming $X,Y$ and $Z$ are sufficiently close to each other, we get that $Lambda$ is nonsingular. Then $X=(Y+Z)[X^T(Y+Z)]^-1$. I can't seem to get further. Perhaps there is some geometric argument?