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

We define a class of $L$-convex-concave subsets of $\mathbb RP^n$, where $L$ is a projective subspace of dimension $l$ in $\mathbb RP^n$. 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 $(\mathbb RP^n)^*$. We discuss a version of Arnold's conjecture for these sets and prove that it is true (or false) for an $L$-convex-concave set and its $L$-dual simultaneously.