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Let $M$ be a compact Riemannian manifold with boundary. How can one recognize whether the manifold can be foliated by strictly convex hypersurfaces?

An exact definition is given below. If the dimension is three or higher, this condition has been shown in some quite recent papers to be sufficient for many interesting positive results in inverse problems. An example of such an inverse problem is this: "If one knows the integrals of a smooth function over all maximal geodesics in $M$, can one reconstruct the function?"

This condition seems to be hard to verify. Are other, more tractable, conditions on the manifold $M$ which imply the foliation condition? In particular, is it sufficient that the manifold is simple? (Simplicity means that any two points on the boundary can be joined by a unique geodesic and the geodesic depends smoothly on its endpoints and the boundary is strictly convex.)

It is known that the foliation condition follows from the absence of focal points, but this assumption is stronger than simplicity. It is also known that there are non-simple manifolds that do have the foliation condition. The definition and some discussion can be found behind the word "recent" above.

I know all the authors of the mentioned papers, and they don't know the answer, but they would very much like to know it. Simplicity is considered a very convenient condition, and it would be significant to know whether it implies the foliation condition.

Definition. Let $(M,g)$ be a compact Riemannian manifold with boundary. We say that $M$ satisfies the foliation condition by strictly convex hypersurfaces if $M$ is equipped with a smooth function $\rho:M\to[0,\infty)$ whose level sets $\Sigma_t=\rho^{-1}(t)$, $t<T$ for some $T>0$ are strictly convex (second fundamental form strictly positive definite) viewed from $\rho^{-1}((t,T))$, $d\rho\neq0$ on these level sets and $M\setminus\bigcup_{0\leq t<T}\Sigma_t$ has empty interior.

If it helps, one can include the additional assumption that $\Sigma_0=\partial M$.

Observation. In dimension two simplicity does imply the foliation condition. A simple surface can be foliated by geodesics, and a small perturbation can make them strictly convex instead of flat. This argument does not seem to generalize to higher dimensions.

Although I am mainly interested in conditions that are stronger than the foliation condition, examples of weaker conditions are also welcome.

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