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Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don't think it matters). Given a closed, convex subset $K \subset X$, there is a result by Rolf Walter that the $\epsilon$-neighbourhood of $K$ (denoted $K_{\epsilon}$) has $C^{1,1}$-regular boundary, i.e. the topological boundary of $K_{\epsilon}$ is a $C^{1,1}$-submanifold of $X$. I was wondering if the regularity can be improved if we allow for more deformations of the convex set. Specifically, I would like to know the following:

${\bf Question:}$ Given a closed, convex subset $K \subset X$ and $\epsilon > 0$, does there exist a convex closed $K'$ such that $K \subset K' \subset K_{\epsilon}$, and $\partial K'$ is a $C^2$-submanifold of $X$?}

Considering that Walter's result goes back to the 70s (Rolf Walter, "Some analytical properties of geodesically convex sets"), this seems like a question whose answer should be known, but I'm struggling to find anything. All methods I could come up with to deform $K$ so that its boundary becomes more smooth while keeping the set convex rely on the second fundamental form, which requires that we make the boundary $C^2$ first.

I should empathize that I do not want to assume that $K$ is compact. (I'm fine assuming that there is a cocompact action by isometries on $K$ though.)

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The answer is "yes" we used a similar argument in our "An optimal lower curvature...". Let me sketch the proof.

I will assume that curvature is negative. The argument will use the existence of cocompact isometric action.

Note that the function $f=\mathrm{dist}^2_K$ is strongly convex in $K_{2\cdot\varepsilon}\backslash K_\varepsilon$.

Let us smooth $f$, applying the argument of Green and Wu [Theorem 2(a) in "$C^∞$ convex functions..."]. Here we have to use the cocompact action; otherwise there might be problems.

Take $K'$ to be set bounded by a level set of smoothed $f$ that lies in $K_{2\cdot\varepsilon}\backslash K_\varepsilon$. This way we get $K'$ with $C^\infty$-smooth boundary.

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  • $\begingroup$ Thanks, this looks very promising. I just want to confirm one thing. Taking a superficial look at the Green-Wu-paper, it seems like their construction yields a smoothed function that is invariant under the cocompact action. (I'm now assuming that the group acts by isometries on $X$, keeps $K$ invariant and acts cocompactly on $K$.) In particular, the set $K'$ we obtain is also invariant under the group action. $\endgroup$ Jul 15, 2021 at 8:13
  • $\begingroup$ @Mathemagica Then things are getting more technical. I may try to think what to do, but please make sure you need it. $\endgroup$ Jul 15, 2021 at 13:02
  • $\begingroup$ It seems that I can make things work without needing any additional invariance of $K'$. No reason to delve into more technicalities. $\endgroup$ Jul 15, 2021 at 15:38
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    $\begingroup$ Just for posterity, since I came across this (very helpful) answer recently: it seems like Theorem 2 (a) of Greene-Wu's ``$C^\infty$ convex functions and manifolds of positive curvature'' might be a better reference? $\endgroup$
    – biringer
    Dec 19, 2022 at 20:48
  • $\begingroup$ @biringer oh yes, thank you very much; I will fix in is a second. $\endgroup$ Dec 19, 2022 at 21:46

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