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We show that for any point $p$ in a closed Riemannian manifold $M$, thereexists at least one point $q\in M$ such that $p$ is critical for the distancefunction from $q$. We also show that such a point $q$ cannot always be reachedwith geodesic loops based at $q$ with midpoint $p$.

Every point in a Riemmanian manifold is critical