\(L\)-convex-concave body in \(\mathbb RP^3\) contains a line (Q1423511)

This paper deals with \(L\)-convex-concave subsets of the projective space \(\mathbb{R}P^n\). The notion of an \(L\)-convex-concave set generalizes the concept of a convex subset of \(\mathbb{R}P^n\) and it can be defined as follows. Let \(L\) be a fixed proper projective subspace of \(\mathbb{R}P^n\). A closed set \(A\subset\mathbb{R}P^n\) is called \(L\)-convex-concave if it satisfies the following conditions: 1) \(A\cap L=\emptyset\); 2) for any projective subspace \(N\) of \(\mathbb{R}P^n\) of dimension \(\dim L+1\) and containing \(L\) the intersection \(A\cap N\) is convex; 3) for any projective subspace \(T\subset L\) of dimension \(\dim L-1\) the complement of the image of \(\Pi(A)\) under the projection \(\Pi:\mathbb{R}P^n\setminus T\to \mathbb{R}P^n/_T\) is an open convex set. In close connection to the Arnold conjecture [\textit{V. Arnold}, Sib. Math. Zh. 29, No. 5(171), 36--47 (1988; Zbl 0668.57003)] about quasi-convex hypersurfaces in \(\mathbb{R}P^n\) the authors formulate the following conjecture: any \(L\)-convex-concave domain \(A\subset\mathbb{R}P^n\) contains a projective subspace of dimension \(n-\dim L-1\). The main result of the paper claims that in case \(n=3\) and \(\dim L=1\) the above conjecture is true. The proof of this result exploits Helly's and Browder's theorems and is guided by the general ideology of Chebyshev's best approximation theory.