L-convex-concave body in ${boldmath{$mathbb{R}P^3$}}$ contains a line

We define a class of L-convex-concave subsets of ${boldmath{$mathbb{R}P^3$}}$, where L is aprojective line in ${boldmath{$mathbb{R}P^3$}}$. These are sets whose sections by any planecontaining L are convex and concavely depend on this plane. Weprove a version of Arnolds conjecture for these sets, namely we provethat each such set contains a line.