# Improving regularity of the boundary of a convex set in Riemannian manifolds

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.)

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.

• 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. Jul 15, 2021 at 8:13
• @Mathemagica Then things are getting more technical. I may try to think what to do, but please make sure you need it. Jul 15, 2021 at 13:02
• It seems that I can make things work without needing any additional invariance of $K'$. No reason to delve into more technicalities. Jul 15, 2021 at 15:38
• 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? Dec 19, 2022 at 20:48
• @biringer oh yes, thank you very much; I will fix in is a second. Dec 19, 2022 at 21:46