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Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.

How much regularity can I obtain for the square of the distance function $$ F(x) := \inf\{d^2(x,y)| y\in N\}? $$

I would be surprised if this extra information had some effect on the answer, but you never know: The embedding is in high codimension; $M$ is dimension $2n+1$ and $N$ is dimension $n-1$.

Thank you very much for every comment, help or reference.

Best wishes Klaus

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welcome to MO! It seems to me that you cannot expect anything more than the obvious, which is local Lipschitz regularity.

First, observe that even with a smooth embedding, there are problems at the some points (where the level hypersurface of the square distance fonction has a "double point"). In the $C^1$ case, you can avoid this in a neighborhood of the submanifold, but I think that if you lack regularity even at one point, then it can happen that these problematic points accumulate to the submanifold.

Take the following examples for a curve in $\mathbb{R}^2$ (or, taking a product, $\mathbb{R}^n$ for any $n$): $$\gamma(t)=\sqrt{|t|}\sin(1/t)$$ It defines a topological embedding of a line (which of course can be made into an embedding of a circle, or a higher-dimensional sphere easily) which only fails to be smooth at one point. It needs a bit of computation, but I think that there is no open set of values for which the square distance function is differentiable.

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