L-convex-concave sets in real projective space and L-duality

We define a class of L-convex-concave subsets of $\Bbb{R}P^n$, where L is a projective subspace of dimension l in $\Bbb{R}P^n$. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-convex-concave set is an $L^*$-convex-concave subset of $(\Bbb RP^n)^*$. We discuss a version of Arnold hypothesis for these sets and prove that it is true (or wrong) for an L-convex-concave set and its L-dual simultaneously.