  We define a class of $L$-convex-concave subsets of $\mathbb{R}P^3$, where $L$
is a projective line in $\mathbb{R}P^3$. These are sets whose sections by any
plane containing $L$ are convex and concavely depend on this plane. We prove a
version of Arnold hypothesis for these sets, namely we prove that each such set
contains a line.
