Foliation of spaces

Question

It turns out that a very important idea to derive properties for a bigger space is to try to foliate the space, derive the same property for each leaf and patch everything up to get the desired conclusion.

My question is: Is it possible/Is there any reference about the possibility of foliating the Euclidean space with boundaries of strictly convex sets (like with boundaries of balls?).

Is there any characterization of a Riemannian manifold so that it can be foliated by leaves made by boundaries of convex sets?

Answer 1

Without specifying additional properties for your leaves this is likely not a very interesting question. An open half-space in Euclidean space is convex and its boundary is a hyperplane. The set of hyperplanes that is orthogonal to a given line trivially foliates Euclidean space. Something tells me that is likely not what you meant?

The added strict convexity requirement doesn't pose much of a challenge either. Firstly, you probably mean that the leaves should be boundaries of sets whose closure is strictly convex, otherwise the above example is still valid. A construction for such a foliation in 2D would be the boundary $y=x^2$ of the strictly convex halfspace $y\geq x^2$ and its vertical translates. In higher dimensions, use this same idea to generate a radially symmetric such foliation (symmetric around one of the coordinate axes).