About a problem of Ulam concerning flat sections of manifolds (Q2640197)

The problem of Ulam mentioned in the title is contained in The Scottish Book [\textit{R. D. Mauldin} (ed.): The Scottish book (1981; Zbl 0485.01013)] as problem 68 and reads as follows: There is given an n- dimensional manifold R (in \({\mathbb{R}}^ n)\) with the property that every section of its boundary by a hyperplane of n-1 dimensions gives an (n-2)- dimensional closed surface (a set homeomorphic to a sphere of this dimension). Prove that R is convex. The purpose of this note is to prove several theorems connected with the problem mentioned above. The author's main results are the following theorems. Theorem 1. Let N be a closed, connected n-manifold topologically embedded in \({\mathbb{R}}^{n+1}\). Suppose that for every n- dimensional hyperplane H that meets N in more than one point, \(H\setminus N\) has exactly two components. Then N is the boundary of a convex \((n+1)\)-body. Theorem 2. Let \(1\leq k\leq n\) and let N be a closed, connected n-manifold topologically embedded in \({\mathbb{R}}^{n+1}\). Suppose that for every k-dimensional hyperplane H that meets the bounded component of \({\mathbb{R}}^{n+1}\setminus N\), \(H\cap N\) has the Čech- cohomology of a (k-1)-sphere. Then N is the boundary of a convex \((n+1)\)- body. Theorem 3. Let \(1\leq k\leq n\) and let K be a compact subset of \({\mathbb{R}}^{n+1}\). Suppose that for every k-dimensional hyperplane H that meets K, \(H\cap K\) is acyclic. Then K is convex. Finally, in a related theorem, sections of a family of parallel hyperplanes with a closed, connected n-manifold topologically embedded in \({\mathbb{R}}^{n+1}\) are considered.